We show that a large class of two-dimensional spinless fermion models exhibit topological superconducting phases characterized by a nonzero Chern number. More specifically, we consider a generic one-band Hamiltonian of spinless fermions that is invariant under both time reversal, T, and a group of rotations and reflections, G, which is either the dihedral point-symmetry group of an underlying lattice, G= Dn, or the orthogonal group of rotations in continuum, G=O (2). Pairing symmetries are classified according to the irreducible representations of T ⊗ G. We prove a theorem that for any two-dimensional representation of this group, a time-reversal symmetry-breaking paired state is energetically favorable. This implies that the ground state of any spinless fermion Hamiltonian in continuum or on a square lattice with a singly connected Fermi surface is always a topological superconductor in the presence of attraction in at least one channel. Motivated by this discovery, we examine phase diagrams of two specific lattice models with nearest-neighbor hopping and attraction on a square lattice and a triangular lattice. In accordance with the general theorem, the former model exhibits only a topological (p+ip) -wave state while the latter shows a doping-tuned quantum phase transition from such state to a nontopological but still exotic f -wave superconductor.