This book presents a comprehensive theory of the correctness of problems related to the Smoluchowski equation, which models the processes of coagulation (merging) of particles in dispersed systems. It examines both spatially homogeneous and heterogeneous problems, proving theorems on global solvability and the correctness of the Cauchy problem. The text describes the effects of transitioning from conservation laws to dissipation relations, highlighting their connection to the emergence of non-smooth features in solutions. Additionally, it proposes approximate methods for solving these problems and provides a solid justification for them. Within the framework of functional solutions, the book outlines an approach to establishing the conditions for the correctness of problems related to Boltzmann-type equations, which encompass the classical Boltzmann equations of kinetic gas theory and the Smoluchowski equation of kinetic coagulation theory. This work is intended for researchers, educators, graduate students, and undergraduates engaged in mathematical studies of models in physical kinetics, colloid chemistry, and biology.