The Lerch-Zeta function is the first monograph to address this topic, which represents a generalization of the classical Riemann and Hurwitz Zeta functions. This book offers a comprehensive analysis of both the analytical and probability theory of the Lerch-Zeta functions. It begins with the classical analytical theory, including the Euler gamma functions and the functional equation, and presents numerous new results, particularly regarding approximate functional equations and their applications, as well as the distribution of zeros. Special attention is given to limit theorems in the sense of weak convergence of probability measures for the Lerch-Zeta function. The universal and functional independence is derived from these limit theorems. This book is beneficial for both researchers and students of analytical and probabilistic number theory and can also be used as a textbook for postgraduate studies.