We discuss an integral equation approach that enables fast computation of the response of nonlinear multi-degree-of-freedom mechanical systems under periodic and quasi-periodic external excitation. The kernel of this integral equation is a Green’s function that we compute explicitly for general mechanical systems. We derive conditions under which the integral equation can be solved by a simple and fast Picard iteration even for non-smooth mechanical systems. The convergence of this iteration cannot be guaranteed for near-resonant forcing, for which we employ a Newton– Raphson iteration instead, obtaining robust convergence. We further show that this integral equation approach can be appended with standard continuation schemes to achieve an additional, significant performance increase over common approaches to computing steady-state response.@en