Solution of functional equations is an old problem of mathematical analysis. Functional equations and their applications are widely studied and can be applied in economics, physics and statistics as well. The results of the book are partly connected to the investigation of functional equations with restricted domains, for example when the restriction of the domain is explicitly given. On the other hand the results are also connected to the examinations, where the equations are satisfied only almost everywhere (with respect to the Lebesgue-measure) on their domain. We also deal with characterization problems of probability theory, such as characterization problems of univariate distributions (for example characterizations of gamma, beta, normal and exponential distributions) as well as characterization problems of conditionally specified bivariate distributions. This book can be helpful for both mathematicians interested in mathematical analysis and researchers interested in characterization problems of probability distributions.