Robot assistants and professional coworkers are becoming a commodity in domestic and industrial settings. In order to enable robots to share their workspace with humans and physically interact with them, fast and reliable handling of possible collisions on the entire robot structure is needed, along with control strategies for safe robot reaction. The primary motivation is the prevention or limitation of possible human injury due to physical contacts. In this survey paper, based on our early work on the subject, we review, extend, compare, and evaluate experimentally model-based algorithms for real-time collision detection, isolation, and identification that use only proprioceptive sensors. This covers the context-independent phases of the collision event pipeline for robots interacting with the environment, as in physical human-robot interaction or manipulation tasks. The problem is addressed for rigid robots first and then extended to the presence of joint/transmission flexibility. The basic physically motivated solution has already been applied to numerous robotic systems worldwide, ranging from manipulators and humanoids to flying robots, and even to commercial products.