Matroids are often introduced as combinatorial abstractions of linear (in)dependence in vector spaces. More formally, a matroid is a (finite and pure) abstract simplicial complex satisfying a certain exchange axiom. It is a geometrical object only formally; however by either introducing the concept of orientation, or by modifying slightly the exchange axiom, it becomes an interesting topological object. In this talk I shall give an overview of topological properties of oriented matroids, such as Mnev's Universality theorem and the Topological Representation Theorem of Folkman and Lawrence. I shall also discuss the so-called symplectic and Lagrangian matroids and their role in the stratification of various Grassmann varieties.