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Modular graph functions are $SL(2,{\mathbb Z})$-invariant functionsassociated with Feynman graphs of a two-dimensional conformal field theory on atorus of modulus $\tau$. For one-loop graphs they reduce to real analyticEisenstein series. We obtain the Fourier series, including the constant andnon-constant Fourier modes, of all two-loop modular graph functions, as well astheir Poincar\'e series with respect to $\Gamma_\infty \backslashPSL(2,{\mathbb Z})$. The Fourier and Poincar\'e series provide the tools tocompute the Petersson inner product of two-loop modular graph functions usingRankin-Selberg-Zagier methods. Modular graph functions which are odd under$\tau \to - \bar \tau$ are cuspidal functions, with exponential decay near thecusp, and exist starting at two loops. Holomorphic subgraph reduction and thesieve algorithm, developed in earlier work, are used to give a lower bound onthe dimension of the space $\mathfrak{A}_w$ of odd two-loop modular graphfunctions of weight $w$. For $w \leq 11$ the bound is saturated and we exhibita basis for $\mathfrak{A}_w$.

Modular graph functions and odd cuspidal functions -- Fourier and Poincar\'e series