This work shows how exponential concentration inequalities for additive functionals of stochastic processes over a finite time interval can be derived from concentration inequalities for martingales. The approach is entirely probabilistic and naturally includes time-inhomogeneous and non-stationary processes as well as initial laws concentrated on a single point. The class of processes studied includes martingales, Markov processes and general square integrable processes. The general approach is complemented by a simple and direct method for martingales, diffusions and discrete-time Markov processes. The method is illustrated by deriving concentration inequalities for the Polyak-Ruppert algorithm, SDEs with time-dependent drift coefficients "contractive at infinity" with both Lipschitz and squared Lipschitz observables, some classical martingales and non-elliptic SDEs.