The goal of this review is to outline some new ideas in the physics of coronal magnetic loops, the fundamental structural elements of the atmospheres of the Sun and flaring stars, which are involved in phenomena such as stellar coronal heating, flare energy release, charged particle acceleration, and the modulation of optical, radio, and X-ray emissions. The Alfvén — Carlqvist view of a coronal loop as an equivalent electric circuit allows a good physical understanding of loop processes. Describing coronal loops as MHD-resonators explains various ways in which flaring emissions from the Sun and stars are modulated, whereas modeling them by magnetic mirror traps allows one to describe the dynamics and emission of high-energy particles. Based on these approaches, loop plasma and fast particle parameters are obtained and models for flare energy release and stellar corona heating are developed.