We characterize the finite-gain L-p stability properties for hybrid dynamical systems. By defining a suitable concept of the hybrid L-p norm, we introduce hybrid storage functions and provide sufficient Lyapunov conditions for the L-p stability of hybrid systems, which cover the well-known continuous-time and discrete-time L-p stability notions as special cases. We then focus on homogeneous hybrid systems and prove a result stating the equivalence among local asymptotic stability of the origin, global exponential stability, existence of a homogeneous Lyapunov function with suitable properties for the hybrid system with no inputs, and input-to-state stability, and we show how these properties all imply L-p stability. Finally, we characterize systems with direct and reverse average dwell-time properties, and establish parallel results for this class of systems. We also make several connections to the existing results on dissipativity properties of hybrid dynamical systems. (C) 2013 Elsevier Ltd. All rights reserved.