Moduli space of sheaves on an algebraic curve is one of the most fundamental objects in modern mathematics. It plays important roles in many areas of mathematics, e.g., algebraic geometry, differential geometry, mathematical physics, number theory, topology, to name a few. In the first part of this talk, I will briefly review the history of the subject. Then I will discuss recent developments about the derived categories of coherent sheaves and the motives of certain moduli spaces of sheaves on algebraic curves.