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An adjoint sensitivity analysis framework to evaluate path-dependent design sensitivities for problems involving inelastic materials and dynamic effects is presented and shown in the context of topology optimization. The overall aim is to present a framework that unifies the sensitivity analyses existing in the literature and provides clear guidelines on how to formulate sensitivity analysis for a wide range of path-dependent system behaviors simulated using finite element analysis (FEA). In particular, the focus is on the identification of proper independent variables for constraint formulation, the overall structure of constraint derivatives arising from discrete FEA equations, and the consistent implementation of sensitivity analysis for diverse problem types. This framework is used to compute sensitivity values for a number of complex problems formulated within FEA for which sensitivity calculations are not available in the literature. The sensitivity values obtained are then rigorously verified using numerical differentiation based on the central-difference method. Problem types include the use of enhanced assumed strain elements, plane-stress constraints, nonlocal elastoplastic-damage formulations, kinematic/isotropic hardening, rate-dependent materials, finite deformations, and dynamics. Finally, topology optimization is carried out using some of the different problem types. KEYWORDSadjoint method, dynamical systems, nonlinear transient systems, path-dependent design sensitivity, topology optimization Int J Numer Methods Eng. 2018;115:1-56. wileyonlinelibrary.com/journal/nme