If M and N are two matroids on the same ground sets, we can speak of the intersection of the two matroids, i.e., the set of sets that are independent both in M and in N. This structure turns out to be quite useful in combinatorics. For example, many problems of a matching in a hypergraph or of a rainbow matching in a graph can be translated into the language of the intersection of two matroids. I will discuss several results and conjectures about the intersection of two matroids, and compare them to what is known about one matroid. Some of the results use topological properties of the intersection of two matroids (viewed as a simplicial complex) which I will describe briefly.