Chow groups give functors from algebraic varieties to abelian groups which are related to (co)homology. However Chow groups frequently contain more information than (co)homology. The construction of Chow groups is easy. Their computation is often difficult. This talk has two aims. First of all it will serve as an introduction to Chow groups which should be accessible to those who have taken a one semester course in Riemann surfaces, two semesters of algebraic topology, and have a passing acquaintance with affine and projective algebraic varieties. (One month in an algebraic geometry course may suffice for the latter.) Given that the next two talks in the algebraic geometry seminar will discuss various aspects of Chow groups, this talk may function as a warm up. The second aim is to introduce Bloch's conjecture on the Chow group of zero dimensional algebraic cycles on a non-singular projective surface. Throughout the talk one may assume that the base field is the complex numbers.