Understanding long‐term coexistence of numerous competing species is a longstanding challenge in ecology. Progress requires determining which processes and species differences are most important for coexistence when multiple processes operate and species differ in many ways. Modern coexistence theory (MCT), formalized by Chesson, holds out the promise of doing that, but empirical applications remain scarce. We argue that MCT\u27s mathematical complexity and subtlety have obscured the simplicity and power of its underlying ideas and hindered applications. We present a general computational approach that extends our previous solution for the storage effect to all of standard MCT\u27s spatial and temporal coexistence mechanisms, and also process‐defined mechanisms amenable to direct study such as resource partitioning, indirect competition, and life history trade‐offs. The main components are a method to partition population growth rates into contributions from different mechanisms and their interactions, and numerical calculations in which some mechanisms are removed and others retained. We illustrate how our approach handles features that have not been analyzed in the standard framework through several case studies: competing diatom species under fluctuating temperature, plant–soil feedbacks in grasslands, facilitation in a beach grass community, and niche differences with independent effects on recruitment, survival and growth in sagebrush steppe
