This paper provides a comprehensive survey of projection-based model reduction methods for parametric dynamical systems. It highlights the challenges and recent advancements in this field, which aims to reduce the computational burden of simulating large-scale dynamical systems by generating reduced models that accurately represent the original system behavior. The paper covers various approaches, including rational interpolation methods, balanced truncation, proper orthogonal decomposition, and greedy algorithms, and discusses their advantages and disadvantages. It also explores the application of parametric model reduction in design, control, optimization, and uncertainty quantification, emphasizing the importance of repeated model evaluations over different parameter values. The paper includes a detailed discussion on error measures, basis computation, parameter sampling, and the construction of parameterized reduced models, providing insights into the state-of-the-art methods and their potential applications.This paper provides a comprehensive survey of projection-based model reduction methods for parametric dynamical systems. It highlights the challenges and recent advancements in this field, which aims to reduce the computational burden of simulating large-scale dynamical systems by generating reduced models that accurately represent the original system behavior. The paper covers various approaches, including rational interpolation methods, balanced truncation, proper orthogonal decomposition, and greedy algorithms, and discusses their advantages and disadvantages. It also explores the application of parametric model reduction in design, control, optimization, and uncertainty quantification, emphasizing the importance of repeated model evaluations over different parameter values. The paper includes a detailed discussion on error measures, basis computation, parameter sampling, and the construction of parameterized reduced models, providing insights into the state-of-the-art methods and their potential applications.