Non-commutative geometry is the geometry of associative algebras, thought of as algebras of functions on "non-commutative spaces". One proposal to make this notion precise, due to Kontsevich and Rosenberg, is to think of the geometry of an associative algebra A as the geometry of the sequence of the (commutative) spaces of all representations of A in n by n matrices, for each n. I will discuss the derived version of this theory, due to Berest, Khachatryan and Ramadoss, the associated notion of representation homology, and the relation with cyclic homology. I will also present some simple examples, such as the algebra of polynomials in two variables, featuring phenomena that are visible in computer experiments, and only partly understood mathematically. (Based on joint work with Y. Berest and A. Ramadoss and with Y. Berest, A. Patotski, A. Ramadoss and T. Willwacher)