This book is intended to provide a fast, interdisciplinary introduction to the basic results of p-adic analysis and its connections with mathematical physics and applications. The book revolves around three topics: (1) p-adic heat equations and ultradiffusion; (2) fundamental solutions and local zeta functions, Riesz kernels, and quadratic forms; (3) Sobolev-type spaces and pseudo-differential evolution equations. These topics are deeply connected with very relevant current research areas. The book includes numerous examples, exercises, and snapshots of several mathematical theories. This book arose from the need to quickly introduce mathematical audience the basic concepts and techniques to do research in p-adic analysis and its connections with mathematical physics and other areas. The book is addressed to a general mathematical audience, which includes computer scientists, theoretical physicists, and people interested in mathematical analysis, PDEs, etc.