  Cell complexes are topological spaces constructed from simple blocks called
cells. They generalize graphs, simplicial complexes, and polyhedral complexes
that form important domains for practical applications. They also provide a
combinatorial formalism that allows the inclusion of complicated relationships
of restrictive structures such as graphs and meshes. In this paper, we propose
\textbf{Cell Complexes Neural Networks (CXNs)}, a general, combinatorial and
unifying construction for performing neural network-type computations on cell
complexes. We introduce an inter-cellular message passing scheme on cell
complexes that takes the topology of the underlying space into account and
generalizes message passing scheme to graphs. Finally, we introduce a unified
cell complex encoder-decoder framework that enables learning representation of
cells for a given complex inside the Euclidean spaces. In particular, we show
how our cell complex autoencoder construction can give, in the special case
\textbf{cell2vec}, a generalization for node2vec.
