diff --git a/src/main/scala/org/scalacheck/Arbitrary.scala b/src/main/scala/org/scalacheck/Arbitrary.scala
index 15774a72b..63d2865d1 100644
--- a/src/main/scala/org/scalacheck/Arbitrary.scala
+++ b/src/main/scala/org/scalacheck/Arbitrary.scala
@@ -268,16 +268,18 @@ object Arbitrary {
   implicit def arbEither[T, U](implicit at: Arbitrary[T], au: Arbitrary[U]): Arbitrary[Either[T, U]] =
     Arbitrary(oneOf(arbitrary[T].map(Left(_)), arbitrary[U].map(Right(_))))
 
-  /** Arbitrary instance of any [[org.scalacheck.util.Buildable]] container (such as lists, arrays,
-   *  streams, etc). The maximum size of the container depends on the size
-   *  generation parameter. */
-  implicit def arbContainer[C[_],T](implicit a: Arbitrary[T], b: Buildable[T,C]
+  /** Arbitrary instance of any [[org.scalacheck.util.Buildable]] container
+   *  (such as lists, arrays, streams, etc). The maximum size of the container
+   *  depends on the size generation parameter. */
+  implicit def arbContainer[C[_],T](implicit
+    a: Arbitrary[T], b: Buildable[T,C], t: C[T] => Traversable[T]
   ): Arbitrary[C[T]] = Arbitrary(containerOf[C,T](arbitrary[T]))
 
-  /** Arbitrary instance of any [[org.scalacheck.util.Buildable2]] container (such as maps, etc).
-   *  The maximum size of the container depends on the size
+  /** Arbitrary instance of any [[org.scalacheck.util.Buildable2]] container
+   *  (such as maps, etc). The maximum size of the container depends on the size
    *  generation parameter. */
-  implicit def arbContainer2[C[_,_],T,U](implicit a: Arbitrary[(T,U)], b: Buildable2[T,U,C]
+  implicit def arbContainer2[C[_,_],T,U](implicit
+    a: Arbitrary[(T,U)], b: Buildable2[T,U,C], t: C[T,U] => Traversable[(T,U)]
   ): Arbitrary[C[T,U]] = Arbitrary(containerOf[C,T,U](arbitrary[(T,U)]))
 
   // Functions //
diff --git a/src/main/scala/org/scalacheck/Gen.scala b/src/main/scala/org/scalacheck/Gen.scala
index 6eb155bb5..e660349cf 100644
--- a/src/main/scala/org/scalacheck/Gen.scala
+++ b/src/main/scala/org/scalacheck/Gen.scala
@@ -266,10 +266,7 @@ object Gen {
         case (rs,g) => g.doApply(p).flatMap(r => rs.map(_ :+ r))
       }
     }
-    val sieve = gs.foldLeft((_:T) => false) { case (s,g) =>
-      x:T => s(x) || g.sieveCopy(x)
-    }
-    g.map(b.fromIterable) // TODO .suchThat(_.forall(sieve))
+    g.map(b.fromIterable)
   }
 
   /** Sequences generators. If any of the given generators fails, the
@@ -280,10 +277,7 @@ object Gen {
         case (rs,g) => g.doApply(p).flatMap(r => rs.map(_ :+ r))
       }
     }
-    val sieve = gs.foldLeft((_:T) => false) { case (s,g) =>
-      x:T => s(x) || g.sieveCopy(x)
-    }
-    g.map(b.fromIterable) // TODO .suchThat(_.forall(sieve))
+    g.map(b.fromIterable)
   }
 
   /** Wraps a generator lazily. The given parameter is only evaluated once,
@@ -348,57 +342,84 @@ object Gen {
 
   //// List Generators ////
 
-  /** Generates a container of any type for which there exists an implicit
+  /** Generates a container of any Traversable type for which there exists an implicit
    *  [[org.scalacheck.util.Buildable]] instance. The elements in the container will
    *  be generated by the given generator. The size of the generated container
    *  is given by `n`. If the given generator fails generating a value, the
-   *  complete container gnerator will also fail. */
-  def containerOfN[C[_],T](n: Int, g: Gen[T])(implicit b: Buildable[T,C]
-  ): Gen[C[T]] = sequence[C,T](Traversable.fill(n)(g))
-
-  /** Generates a container of any type for which there exists an implicit
-   *  [[org.scalacheck.util.Buildable]] instance. The elements in the container
-   *  will be generated by the given generator. The size of the container is
-   *  bounded by the size parameter used when generating values. */
-  def containerOf[C[_],T](g: Gen[T])(implicit b: Buildable[T,C]): Gen[C[T]] =
-    sized(size => choose(0,size).flatMap(containerOfN[C,T](_,g)))
+   *  complete container generator will also fail. */
+  def containerOfN[C[_],T](n: Int, g: Gen[T])(implicit
+    evb: Buildable[T,C], evt: C[T] => Traversable[T]
+  ): Gen[C[T]] =
+    sequence[C,T](Traversable.fill(n)(g)) suchThat { c =>
+      c.size == n && c.forall(g.sieveCopy)
+    }
 
-  /** Generates a non-empty container of any type for which there exists an
+  /** Generates a container of any Traversable type for which there exists an
    *  implicit [[org.scalacheck.util.Buildable]] instance. The elements in the
    *  container will be generated by the given generator. The size of the
    *  container is bounded by the size parameter used when generating values. */
-  def nonEmptyContainerOf[C[_],T](g: Gen[T])(implicit b: Buildable[T,C]): Gen[C[T]] =
-    sized(size => choose(1,size).flatMap(containerOfN[C,T](_,g)))
+  def containerOf[C[_],T](g: Gen[T])(implicit
+    evb: Buildable[T,C], evt: C[T] => Traversable[T]
+  ): Gen[C[T]] =
+    sized(s => choose(0,s).flatMap(containerOfN[C,T](_,g))) suchThat { c =>
+      c.forall(g.sieveCopy)
+    }
 
-  /** Generates a non-empty container of any type for which there exists an
-   *  implicit [[org.scalacheck.util.Buildable]] instance. The elements in the
-   *  container will be generated by the given generator. The size of the
-   *  container is bounded by the size parameter used when generating values. */
-  @deprecated("Use Gen.nonEmptyContainerOf instead", "1.11.0")
-  def containerOf1[C[_],T](g: Gen[T])(implicit b: Buildable[T,C]): Gen[C[T]] =
-    nonEmptyContainerOf[C,T](g)
+  /** Generates a non-empty container of any Traversable type for which there
+   *  exists an implicit [[org.scalacheck.util.Buildable]] instance. The
+   *  elements in the container will be generated by the given generator. The
+   *  size of the container is bounded by the size parameter used when
+   *  generating values. */
+  def nonEmptyContainerOf[C[_],T](g: Gen[T])(implicit
+    evb: Buildable[T,C], evt: C[T] => Traversable[T]
+  ): Gen[C[T]] =
+    sized(s => choose(1,s).flatMap(containerOfN[C,T](_,g))) suchThat { c =>
+      c.size > 0 && c.forall(g.sieveCopy)
+    }
 
-  /** Generates a container of any type for which there exists an implicit
-   *  [[org.scalacheck.util.Buildable2]] instance. The elements in the container will
-   *  be generated by the given generator. The size of the generated container
-   *  is given by `n`. If the given generator fails generating a value, the
-   *  complete container gnerator will also fail. */
-  def containerOfN[C[_,_],T,U](n: Int, g: Gen[(T,U)])(implicit b: Buildable2[T,U,C]
-  ): Gen[C[T,U]] = sequence[C,T,U](Traversable.fill(n)(g))
+  /** Generates a non-empty container of any Traversable type for which there
+   *  exists an implicit [[org.scalacheck.util.Buildable]] instance. The
+   *  elements in the container will be generated by the given generator. The
+   *  size of the container is bounded by the size parameter used when
+   *  generating values. */
+  @deprecated("Use Gen.nonEmptyContainerOf instead", "1.11.0")
+  def containerOf1[C[_],T](g: Gen[T])(implicit
+    evb: Buildable[T,C], evt: C[T] => Traversable[T]
+  ): Gen[C[T]] = nonEmptyContainerOf[C,T](g)
+
+  /** Generates a container of any Traversable type for which there exists an
+   *  implicit [[org.scalacheck.util.Buildable2]] instance. The elements in
+   *  the container will be generated by the given generator. The size of the
+   *  generated container is given by `n`. If the given generator fails
+   *  generating a value, the complete container generator will also fail. */
+  def containerOfN[C[_,_],T,U](n: Int, g: Gen[(T,U)])(implicit
+    evb: Buildable2[T,U,C], evt: C[T,U] => Traversable[(T,U)]
+  ): Gen[C[T,U]] =
+    sequence[C,T,U](Traversable.fill(n)(g)).suchThat { c =>
+      c.size == n && c.forall(g.sieveCopy)
+    }
 
-  /** Generates a container of any type for which there exists an implicit
-   *  <code>Buildable2</code> instance. The elements in the container will
-   *  be generated by the given generator. The size of the container is
-   *  bounded by the size parameter used when generating values. */
-  def containerOf[C[_,_],T,U](g: Gen[(T,U)])(implicit b: Buildable2[T,U,C]): Gen[C[T,U]] =
-    sized(size => for(n <- choose(0,size); c <- containerOfN[C,T,U](n,g)) yield c)
+  /** Generates a container of any Traversable type for which there exists
+   *  an implicit <code>Buildable2</code> instance. The elements in the
+   *  container will be generated by the given generator. The size of the
+   *  container is bounded by the size parameter used when generating values. */
+  def containerOf[C[_,_],T,U](g: Gen[(T,U)])(implicit
+    evb: Buildable2[T,U,C], evt: C[T,U] => Traversable[(T,U)]
+  ): Gen[C[T,U]] =
+    sized(s => choose(0,s).flatMap(containerOfN[C,T,U](_,g))) suchThat { c =>
+      c.forall(g.sieveCopy)
+    }
 
   /** Generates a non-empty container of any type for which there exists an
    *  implicit <code>Buildable2</code> instance. The elements in the container
    *  will be generated by the given generator. The size of the container is
    *  bounded by the size parameter used when generating values. */
-  def nonEmptyContainerOf[C[_,_],T,U](g: Gen[(T,U)])(implicit b: Buildable2[T,U,C]): Gen[C[T,U]] =
-    sized(size => for(n <- choose(1,size); c <- containerOfN[C,T,U](n,g)) yield c)
+  def nonEmptyContainerOf[C[_,_],T,U](g: Gen[(T,U)])(implicit
+    evb: Buildable2[T,U,C], evt: C[T,U] => Traversable[(T,U)]
+  ): Gen[C[T,U]] =
+    sized(s => choose(1,s).flatMap(containerOfN[C,T,U](_,g))) suchThat { c =>
+      c.size > 0 && c.forall(g.sieveCopy)
+    }
 
   /** Generates a list of random length. The maximum length depends on the
    *  size parameter. This method is equal to calling
@@ -482,16 +503,18 @@ object Gen {
 
   /* Generates a string that starts with a lower-case alpha character,
    * and only contains alphanumerical characters */
-  def identifier: Gen[String] = for {
+  def identifier: Gen[String] = (for {
     c <- alphaLowerChar
     cs <- listOf(alphaNumChar)
-  } yield (c::cs).mkString // TODO suchThat
+  } yield (c::cs).mkString).suchThat(_.forall(c => c.isLetter || c.isDigit))
 
   /* Generates a string of alpha characters */
-  def alphaStr: Gen[String] = listOf(alphaChar).map(_.mkString) // TODO suchThat
+  def alphaStr: Gen[String] =
+    listOf(alphaChar).map(_.mkString).suchThat(_.forall(_.isLetter))
 
   /* Generates a string of digits */
-  def numStr: Gen[String] = listOf(numChar).map(_.mkString) // TODO suchThat
+  def numStr: Gen[String] =
+    listOf(numChar).map(_.mkString).suchThat(_.forall(_.isDigit))
 
 
   //// Number Generators ////
diff --git a/src/main/scala/org/scalacheck/util/Buildable.scala b/src/main/scala/org/scalacheck/util/Buildable.scala
index fac8b0a50..eb959ee6a 100644
--- a/src/main/scala/org/scalacheck/util/Buildable.scala
+++ b/src/main/scala/org/scalacheck/util/Buildable.scala
@@ -18,7 +18,6 @@ trait Buildable[T,C[_]] {
     b ++= it
     b.result()
   }
-  // TODO def toIterable
 }
 
 trait Buildable2[T,U,C[_,_]] {
diff --git a/src/test/scala/org/scalacheck/util/BuildableSpecification.scala b/src/test/scala/org/scalacheck/util/BuildableSpecification.scala
index f44bf474e..4f23462e1 100644
--- a/src/test/scala/org/scalacheck/util/BuildableSpecification.scala
+++ b/src/test/scala/org/scalacheck/util/BuildableSpecification.scala
@@ -13,8 +13,10 @@ package util
 import collection._
 
 object BuildableSpecification {
-  def container[C[_]](implicit ev: Buildable[String, C]) =
-    Gen.containerOf[C, String](Gen.alphaStr)
+  def container[C[_]](implicit
+    evb: Buildable[String, C],
+    evt: C[String] => Traversable[String]
+  ) = Gen.containerOf[C, String](Gen.alphaStr)
 
   implicit val listGen: Gen[List[String]] = container[List]
 
@@ -41,4 +43,4 @@ object BuildableSpecification {
   implicit val iterableGen: Gen[immutable.Iterable[String]] = container[immutable.Iterable]
 
   implicit val trieIteratorGen: Gen[immutable.Queue[String]] = container[immutable.Queue]
-}
\ No newline at end of file
+}