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Since early 70's, mixed finite elements have been the object of a wide and deep study by the mathematical and engineering communities. The fundamental role of this method for many application fields has been worldwide recognized and their use has been introduced in several commercial codes. An important feature of mixed finite elements is the interplay between theory and application; from one side, many schemes used for real life simulations have been casted in a rigorous framework, from the other side, the theoretical analysis makes it possible to design new schemes or to improve existing ones based on their mathematical properties. Indeed, due to the compatibility conditions required by the discretization spaces in order to provide stable schemes, simple minded approximations generally do not work and the design of suitable stabilizations gives rise to challenging mathematical problems. With this course we pursue two main goals. The first one is to review the rigorous setting of mixed finite elements and to revisit it after more than 30 years of practice; this compounds in developing a detailed a priori and a posteriori analysis for the h, p, and hp versions. The second one consists in showing some examples of possible applications of the method. The applications range from traditional ones, like fluid-dynamics or elasticity, to more recent fields, like electromagnetism where the research is particularly active. We expect in the audience Ph.D. students and researchers in numerical analysis and related fields, who aim at improving their knowledge about the topics of the course, and, more generally, everybody who would like to interact with major experts on mixed finite elements and learn about the latest related developments.