diff --git a/test/test_edge_index.py b/test/test_edge_index.py
index 37fe4609c5b7..7a396c6a8f6e 100644
--- a/test/test_edge_index.py
+++ b/test/test_edge_index.py
@@ -88,6 +88,7 @@ def test_identity(dtype, device, is_undirected):
     adj = EdgeIndex([[0, 1, 1, 2], [1, 0, 2, 1]], sparse_size=(3, 3), **kwargs)
 
     out = EdgeIndex(adj)
+    print(out._data)
     assert out.data_ptr() == adj.data_ptr()
     assert out.dtype == adj.dtype
     assert out.device == adj.device
diff --git a/torch_geometric/edge_index.py b/torch_geometric/edge_index.py
index dfe72ba7fac7..d418a51a5731 100644
--- a/torch_geometric/edge_index.py
+++ b/torch_geometric/edge_index.py
@@ -138,8 +138,8 @@ def assert_symmetric(size: Tuple[Optional[int], Optional[int]]) -> None:
 
 def assert_sorted(func: Callable) -> Callable:
     @functools.wraps(func)
-    def wrapper(*args: Any, **kwargs: Any) -> Any:
-        if not args[0].is_sorted:
+    def wrapper(self: 'EdgeIndex', *args: Any, **kwargs: Any) -> Any:
+        if not self.is_sorted:
             cls_name = args[0].__class__.__name__
             raise ValueError(
                 f"Cannot call '{func.__name__}' since '{cls_name}' is not "
@@ -222,7 +222,7 @@ class EdgeIndex(Tensor):
     # The size of the underlying sparse matrix:
     _sparse_size: Tuple[Optional[int], Optional[int]] = (None, None)
 
-    # Whether the `edge_index` represented is non-sorted (`None`), or sorted
+    # Whether the `edge_index` representation is non-sorted (`None`), or sorted
     # based on row or column values.
     _sort_order: Optional[SortOrder] = None
 
@@ -344,6 +344,7 @@ def __new__(
         out._indptr = indptr
 
         if isinstance(data, cls):  # If passed `EdgeIndex`, inherit metadata:
+            out._data = data._data
             out._T_perm = data._T_perm
             out._T_index = data._T_index
             out._T_indptr = data._T_indptr
@@ -1102,6 +1103,17 @@ def sparse_narrow(
             edge_index._indptr = colptr
             return edge_index
 
+    def to_vector(self) -> Tensor:
+        r"""Converts :class:`EdgeIndex` into a one-dimensional index
+        vector representation.
+        """
+        num_rows, num_cols = self.get_sparse_size()
+
+        if num_rows * num_cols > torch_geometric.typing.MAX_INT64:
+            raise ValueError("'to_vector()' will result in an overflow")
+
+        return self._data[0] * num_rows + self._data[1]
+
     # PyTorch/Python builtins #################################################
 
     def __tensor_flatten__(self) -> Tuple[List[str], Tuple[Any, ...]]:
@@ -1228,17 +1240,6 @@ def _clear_metadata(self) -> 'EdgeIndex':
         self._cat_metadata = None
         return self
 
-    def to_vector(self) -> Tensor:
-        r"""Converts :class:`EdgeIndex` into a one-dimensional index
-        vector representation.
-        """
-        num_rows, num_cols = self.get_sparse_size()
-
-        if num_rows * num_cols > torch_geometric.typing.MAX_INT64:
-            raise ValueError("'to_vector()' will result in an overflow")
-
-        return self._data[0] * num_rows + self._data[1]
-
 
 class SortReturnType(NamedTuple):
     values: EdgeIndex
diff --git a/torch_geometric/index.py b/torch_geometric/index.py
index 6c05e330ac3c..e5cf6b2d65b6 100644
--- a/torch_geometric/index.py
+++ b/torch_geometric/index.py
@@ -1,8 +1,15 @@
-from typing import Optional
+import functools
+from typing import Any, Callable, Dict, List, NamedTuple, Optional, Type, Union
 
 import torch
 from torch import Tensor
 
+from torch_geometric.typing import INDEX_DTYPES
+
+aten = torch.ops.aten
+
+HANDLED_FUNCTIONS: Dict[Callable, Callable] = {}
+
 
 def ptr2index(ptr: Tensor, output_size: Optional[int] = None) -> Tensor:
     index = torch.arange(ptr.numel() - 1, dtype=ptr.dtype, device=ptr.device)
@@ -15,3 +22,186 @@ def index2ptr(index: Tensor, size: Optional[int] = None) -> Tensor:
 
     return torch._convert_indices_from_coo_to_csr(
         index, size, out_int32=index.dtype != torch.int64)
+
+
+class CatMetadata(NamedTuple):
+    nnz: List[int]
+    dim_size: List[Optional[int]]
+    is_sorted: List[bool]
+
+
+def implements(torch_function: Callable) -> Callable:
+    r"""Registers a :pytorch:`PyTorch` function override."""
+    @functools.wraps(torch_function)
+    def decorator(my_function: Callable) -> Callable:
+        HANDLED_FUNCTIONS[torch_function] = my_function
+        return my_function
+
+    return decorator
+
+
+def assert_valid_dtype(tensor: Tensor) -> None:
+    if tensor.dtype not in INDEX_DTYPES:
+        raise ValueError(f"'Index' holds an unsupported data type "
+                         f"(got '{tensor.dtype}', but expected one of "
+                         f"{INDEX_DTYPES})")
+
+
+def assert_one_dimensional(tensor: Tensor) -> None:
+    if tensor.dim() != 1:
+        raise ValueError(f"'Index' needs to be one-dimensional "
+                         f"(got {tensor.dim()} dimensions)")
+
+
+def assert_contiguous(tensor: Tensor) -> None:
+    if not tensor.is_contiguous():
+        raise ValueError("'Index' needs to be contiguous. Please call "
+                         "`index.contiguous()` before proceeding.")
+
+
+def assert_sorted(func: Callable) -> Callable:
+    @functools.wraps(func)
+    def wrapper(self: 'Index', *args: Any, **kwargs: Any) -> Any:
+        if not self.is_sorted:
+            cls_name = self.__class__.__name__
+            raise ValueError(
+                f"Cannot call '{func.__name__}' since '{cls_name}' is not "
+                f"sorted. Please call `{cls_name}.sort()` first.")
+        return func(*args, **kwargs)
+
+    return wrapper
+
+
+class Index(Tensor):
+    r"""TODO."""
+    # See "https://pytorch.org/docs/stable/notes/extending.html"
+    # for a basic tutorial on how to subclass `torch.Tensor`.
+
+    # The underlying tensor representation:
+    _data: Tensor
+
+    # The size of the underlying sparse vector, e.g. `_data.max() + 1` :
+    _dim_size: Optional[int] = None
+
+    # Whether the `index` representation is sorted:
+    _is_sorted: bool = False
+
+    # A cache for its compressed representation:
+    _indptr: Optional[Tensor] = None
+
+    # Whenever we perform a concatenation of indices, we cache the original
+    # metadata to be able to reconstruct individual indices:
+    _cat_metadata: Optional[CatMetadata] = None
+
+    @staticmethod
+    def __new__(
+        cls: Type,
+        data: Any,
+        *args: Any,
+        dim_size: Optional[int] = None,
+        is_sorted: bool = False,
+        **kwargs: Any,
+    ) -> 'Index':
+        if not isinstance(data, Tensor):
+            data = torch.tensor(data, *args, **kwargs)
+        elif len(args) > 0:
+            raise TypeError(
+                f"new() received an invalid combination of arguments - got "
+                f"(Tensor, {', '.join(str(type(arg)) for arg in args)})")
+        elif len(kwargs) > 0:
+            raise TypeError(f"new() received invalid keyword arguments - got "
+                            f"{set(kwargs.keys())})")
+
+        assert isinstance(data, Tensor)
+
+        indptr: Optional[Tensor] = None
+
+        if isinstance(data, cls):  # If passed `Index`, inherit metadata:
+            indptr = data._indptr
+            dim_size = dim_size or data.dim_size()
+            is_sorted = is_sorted or data.is_sorted
+
+        assert_valid_dtype(data)
+        assert_one_dimensional(data)
+        assert_contiguous(data)
+
+        out = Tensor._make_wrapper_subclass(  # type: ignore
+            cls,
+            size=data.size(),
+            strides=data.stride(),
+            dtype=data.dtype,
+            device=data.device,
+            layout=data.layout,
+            requires_grad=False,
+        )
+        assert isinstance(out, Index)
+
+        # Attach metadata:
+        out._data = data
+        out._dim_size = dim_size
+        out._is_sorted = is_sorted
+        out._indptr = indptr
+
+        if isinstance(data, cls):
+            out._data = data._data
+
+            # Reset metadata if cache is invalidated:
+            if dim_size is not None and dim_size != data.dim_size():
+                out._indptr = None
+
+        return out
+
+    # Validation ##############################################################
+
+    def validate(self) -> 'Index':
+        raise NotImplementedError
+
+    # Properties ##############################################################
+
+    @property
+    def dim_size(self) -> Optional[int]:
+        raise NotImplementedError
+
+    @property
+    def is_sorted(self) -> bool:
+        raise NotImplementedError
+
+    # Cache Interface #########################################################
+
+    def get_dim_size(self) -> int:
+        raise NotImplementedError
+
+    def dim_resize_(self, dim_size: Optional[int]) -> 'Index':
+        raise NotImplementedError
+
+    @assert_sorted
+    def get_indptr(self) -> Tensor:
+        raise NotImplementedError
+
+    def fill_cache_(self) -> 'Index':
+        raise NotImplementedError
+
+    # Methods #################################################################
+
+    def share_memory_(self) -> 'Index':
+        self._data.share_memory_()
+        if self._indptr is not None:
+            self._indptr.share_memory_()
+        return self
+
+    def is_shared(self) -> bool:
+        return self._data.is_shared()
+
+    def as_tensor(self) -> Tensor:
+        r"""Zero-copies the :class:`Index` representation back to a
+        :class:`torch.Tensor` representation.
+        """
+        return self._data
+
+    def dim_narrow(self, start: Union[int, Tensor], length: int) -> 'Index':
+        raise NotImplementedError
+
+
+def sort(self) -> None:
+    # TODO MOVE BEHIND TORCH DISPATCH
+    raise NotImplementedError