I will discuss topological classification of gapped many-body Hamiltonians and their ground states in dimension d. In general, the problem is too hard as it includes diverse phenomena such as degenerate ground states on the torus, anyons, and fractons. There is, however, hope to fully understand short-range entangled, or "invertible" systems, which do not have the aforementioned features but may possess gapless edge modes. In the case of non-interacting fermions, the topological classification is based on K-theory. I will argue that general invertible systems are described by some homotopy spectrum (or equivalently, generalized cohomology theory). However, the exact answer is not yet known.