Legged mobility has long been among key research areas in mobile robotics. In this context, accurate dynamic models of locomotory behaviors provide tools that are useful both in understanding biological systems as well as constructing robots and controllers to realize these behaviors. In this talk, I will focus on the latter, using spring-mass models that have been instrumental in the understanding and artificial realization of running behaviors. I will first describe our work in finding approximate analytic solutions for spring-mass models of running, which possess otherwise non-integrable stance dynamics. I will then show different applications of these solutions, including adaptive control, state estimation and footstep planning for planar running. Subsequently, I will describe a new method for energy regulation through virtual tuning of damping properties for such systems, towards a level of energy and power efficiency that has not been possible with previous methods.