Combinatorial algebraic geometry seeks to understand combinatorial structures such as graphs, matroids, and polytopes through geometry of algebraic varieties and vice versa. Recent years have seen developments in CAG, particularly in the context of matroid theory and more generally in 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.