Recent years have seen developments in Combinatorial Algebraic Geometry, particularly in the context of matroid theory and more generally in the context of tropical geometry. This course aims to go over some major fronts of these developments. Relevant topics include: Baker-Bowler theory of matroids, Foundations of matroids, Chow rings of matroids and wonderful compactifications, Tropical Hodge theory, intersection cohomology of a matroid, Tautological classes of matroids, Lorentzian and stable polynomials.