The classical Kazhdan-Lusztig polynomials have brought a wealth of interesting mathematics for decades. In this talk we will survey a new invariant of a matroid which closely resembles these famous polynomials. The matroid Kazhdan-Lusztig polynomials have a striking similarity to the classical case. They are intersection cohomology Poincare polynomials of an interesting algebraic variety: the reciprocal plane. The Catalan numbers and other interesting combinatorial numbers turn up as coefficients of these polynomials for certain families of matroids. One exciting feature of these polynomials is that at this time there is very little known about their structure and lots of open conjectures. We will highlight some of these conjectures together with some of the examples which provided the motivation.