This book is the first of two volumes on random motions in Markov andsemi-Markov random environments. This first volume focuses onhomogenous random motions.This volume consists of two parts, the first describing the basic conceptsand methods that have been developed for random evolutions. Thesemethods are the foundational tools used in both volumes, and thisdescription includes many results in potential operators.Sometechniques to find closed-form expressions in relevant applications arealso presented.The second part deals with asymptotic results and presents a variety ofapplications, including random motion with different types ofboundaries, the reliability of storage systems and solutions of partialdifferential equations with constant coefficients, using commutativealgebra techniques. It also presents an alternative formulation to theBlack-Scholes formula in finance, fading evolutions and telegraphprocesses, including jump telegraph processes and the estimation of thenumber of level crossings for telegraph processes.