Complex phenomena in different branches of human knowledge (such as e.g. Physics, Engineering, Biological and Social Sciences) can be simulated by sophisticated numerical methods, providing a high-fidelity description of reality. However, often these simulation techniques turn out to be expensive or slow, particularly when responses are needed in large numbers or in real-time. Reduced order methods have been devised to surrogate the accurate but expensive simulation models, by producing results that are accurate enough but cheaper to obtain. Proper Orthogonal Decompositions (POD), or Reduced Basis Methods (RBM) are examples of such techniques. More recently, the use of Machine Learning (ML) techniques has been advocated as a viable alternative to more classical strategies, particularly in the construction of nonlinear models. The course is an introduction to Reduced-Order Methods on the one hand, and to Machine Learning on the other hand, both in their theoretical aspects and in their practical implementation. They will be combined to produce efficient solution strategies, which will be applied to treat representative mathematical models described by partial differential equations. Applications will range from parametric boundary-value problems to uncertainty quantification or optimal control problems. The course provides essential knowledge on surrogate simulation models, which students could exploit in many scientific and industrial contexts during their future careers.