The primary mathematical focus of this monograph is on regular multi-frequency systems of ordinary differential equations with slow and fast phase variables, where resonance relationships between fundamental frequencies may occur. It presents a classification of resonant systems based on the concept of whether solutions become stuck or not in the vicinity of resonance points. A constructive perturbation theory is developed, employing the principle of averaging (smoothing), Poincaré-type asymptotic representations, and iterative variations of the classical Lyapunov-Poincaré method. In most cases, it is possible to construct approximate solutions for multi-frequency systems in either analytical or numerically-analytical forms with any desired accuracy concerning the small parameter. The text also addresses some general issues related to computer technologies in the asymptotic theory of differential equations and constructive methods for deriving first and higher-order approximations. The effectiveness of the developed asymptotic theory is illustrated through various problems in applied nonlinear analysis. This book is intended for specialists in nonlinear analysis, mathematical modeling, and computational mathematics.