•The concept of FORM importance measures are generalized to handle challenging problems.•GRIM overcome challenges in problems with multiple critical subdomains or large curvature.•Using regional participation factors, the relative importance is evaluated per each critical region.•GRIMs are computed using a Gaussian mixture model fitted to the density in the failure domain.•Various numerical examples successfully demonstrate merits and applicability of GRIMs. In structural reliability analysis, it is often desirable to evaluate the relative contributions of random variables to the variability of the limit-state function in the failure domain. Based on the relative contributions, one can effectively reduce the dimension of the reliability problem or obtain useful insight and information. However, existing reliability importance measures, which are available as a by-product of reliability analysis by first-order reliability method (FORM), may not capture the contributions of random variables accurately when the limit-state surface shows a large curvature around the design point or multiple critical subdomains exist in the failure domain. To address the issue, this paper proposes a Generalized Reliability Importance Measure (GRIM) that can deal with multiple critical failure regions, large curvatures of limit-state surfaces and the correlation between the input random variables. By introducing Gaussian mixture and the regional participation factor, the failure domain is divided into several subdomains, and the relative contributions of random variables in each critical domain are evaluated. To facilitate the computations of GRIMs, the cross-entropy-based adaptive importance sampling technique (CE-AIS-GM) is employed to identify the locations of critical subdomains, their relative contributions and corresponding importance vectors. Eight numerical examples covering a variety of component and system reliability problems demonstrate the proposed method and its merits. The test results confirm robust performance against the number of important regions or the dimension. The proposed GRIMs and computational procedure are expected to provide more reliable measures for a wide range of component and system reliability problems. The supporting source code and data are available for download at https://github.com/TyongKim/GRIM.