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Topological insulators and superconductors are fermionic symmetry-protected topological phases. Namely, they are topologically distinct from trivial states of matter in the presence of some symmetries. For example, topological insulators are topological only in the presence of time-reversal symmetry. Topological properties of such fermionic symmetry-protected topologically phases are commonly discussed by using single-particle Bloch wave functions. While such single-particle formulations have been tremendously successful, there still remains a question to address — the effect of interactions. In this talk, I will describe a fully many-body formulation of topological invariants for various symmetry-protected topological phases of matter, which does not refer to single particle wave functions at all.