Complex phenomena across various fields (e.g. Physics, Engineering, Biological and Social Sciences) can be simulated using advanced numerical methods to achieve high-fidelity representations of reality. However, these simulation techniques are often computational expensive or slow, particularly when large numbers of responses or real-time results are required. To address this, Reduced-Order methods have been developed as efficient alternatives to these accurate but expensive simulation models, by producing accurate enough results at a lower computational cost. Examples of such techniques include Proper Orthogonal Decompositions (POD) and Reduced Basis Methods (RBM). Recently, Machine Learning (ML), in particular with Neural Networks, has emerged as a viable alternative to classical strategies, particularly for constructing nonlinear models. This course introduces Reduced-Order Methods with both the traditional Galerkin and the new Machine Learning approach, covering their theoretical aspects and practical implementations. By combining these techniques, the course aims to develop efficient solution strategies for mathematical models described by Partial Differential Equations (PDEs). Applications will include parametric boundary-value problems, uncertainty quantification, and optimal control problems. The course equips students with essential knowledge of surrogate simulation models, which they can apply in various scientific and industrial contexts throughout their careers.