Since their first introduction in 1935, matroids have always been considered a great combinatorial tool for branches of applied mathematics like Coding Theory and Optimization. However, in the last years, a deep interplay between Matroid Theory, Algebraic Topology and Algebraic Geometry has been found: the Combinatorics of matroids allows us to fully describe the cohomology of some interesting varieties (the complement of an arrangement of hyperplanes, the wonderful model of De Concini-Procesi, the reciprocal plane); their Geometry led the way to solve long-standing combinatorial conjectures like the Heron-Rota-Welsh Conjecture and the Top-Heavy conjecture. In this talk, we will present Kazhdan-Lusztig polynomials for matroids, outlining the geometric setting in which they were first introduced in 2016. Many of their properties are still only conjectured, since their definition is given through an intricate recursion. We will show how one can partially solve these conjectures for wide classes of matroids using just combinatorial tools. This is a joint work with L. Ferroni and G.D. Nasr.