This review presents an overview of critical phenomena and renormalization-group (RG) theory, focusing on the critical behavior of spin systems at equilibrium. It covers the Ising and general $ O(N) $-symmetric universality classes, including the $ N \rightarrow 0 $ limit that describes self-avoiding walks. The paper discusses critical exponents, the equation of state, amplitude ratios, and the two-point function of the order parameter for systems in three and two dimensions. It also addresses crossover phenomena, particularly in systems with medium-range interactions, and explores more complex Landau-Ginzburg-Wilson Hamiltonians for magnetic and structural phase transitions. The review includes results for the tetragonal Landau-Ginzburg-Wilson Hamiltonian with three quartic couplings, presenting the six-loop perturbative series for the $ \beta $-functions. Additionally, it considers a Hamiltonian with symmetry $ O(n_1) \oplus O(n_2) $ relevant for multicritical phenomena. The paper outlines the theoretical framework of critical phenomena, including scaling relations, critical indices, and the behavior of thermodynamic quantities near the critical point. It also discusses the scaling behavior of the free energy and equation of state, as well as the critical crossover between the Gaussian and Wilson-Fisher fixed points. The review emphasizes the universality of critical behavior across different systems and the importance of critical exponents, amplitude ratios, and scaling functions in understanding phase transitions.This review presents an overview of critical phenomena and renormalization-group (RG) theory, focusing on the critical behavior of spin systems at equilibrium. It covers the Ising and general $ O(N) $-symmetric universality classes, including the $ N \rightarrow 0 $ limit that describes self-avoiding walks. The paper discusses critical exponents, the equation of state, amplitude ratios, and the two-point function of the order parameter for systems in three and two dimensions. It also addresses crossover phenomena, particularly in systems with medium-range interactions, and explores more complex Landau-Ginzburg-Wilson Hamiltonians for magnetic and structural phase transitions. The review includes results for the tetragonal Landau-Ginzburg-Wilson Hamiltonian with three quartic couplings, presenting the six-loop perturbative series for the $ \beta $-functions. Additionally, it considers a Hamiltonian with symmetry $ O(n_1) \oplus O(n_2) $ relevant for multicritical phenomena. The paper outlines the theoretical framework of critical phenomena, including scaling relations, critical indices, and the behavior of thermodynamic quantities near the critical point. It also discusses the scaling behavior of the free energy and equation of state, as well as the critical crossover between the Gaussian and Wilson-Fisher fixed points. The review emphasizes the universality of critical behavior across different systems and the importance of critical exponents, amplitude ratios, and scaling functions in understanding phase transitions.