The nodal set of a Laplacian eigenfunction forms a partition of the underlying manifold or graph. Another natural partition is based on the gradient vector field of the eigenfunction (on a manifold) or on the extremal points of the eigenfunction (on a graph). The submanifolds (or subgraphs) of this partition are called Neumann domains. We present the main results concerning Neumann domains on manifolds and on graphs. We compare manifolds to graphs and relate the Neumann domain results on each of them to the nodal domain study. The talk is based on joint works with Lior Alon, Michael Bersudsky, Sebastian Egger, David Fajman and Alexander Taylor.