This graduate textbook accompanies the reader starting from the basics in the theory of elliptic problems to recent research topics concerning the asymptotic theory in cylindrical domains where many physical problems are settled. It is designed as a resource for graduates and researchers in applied mathematics and for engineers. Many physical problems are meaningfully formulated in a cylindrical domain. When the size of the cylinder goes to infinity, the solutions, under certain symmetry conditions, are expected to be identical in every cross-section of the domain. The proof of this, however, is sometimes difficult and almost never given in the literature. This book aims to partially fill this gap by providing proofs of the asymptotic behaviour of solutions to various important cases of linear and non-linear problems in the theory of elliptic and parabolic partial differential equations. Many results presented here are original and have not been published elsewhere. They are included with the aim of motivating and enabling the reader to apply the theory to other problems in partial differential equations.