*Abstract:* I will give an introduction to the main themes of my Ph.D. thesis, in which a theory of Chow groups for tensor triangulated categories is developed. First, I will discuss some examples of tensor triangulated categories from representation theory (stable module categories) and algebraic geometry (derived categories of perfect complexes). Then I will recall the concept of Chow groups of an algebraic variety and show how "tensor triangular geometry" can be used to generalize these invariants to a categorical setting. As a small application, I will show that the stable module category of the Klein four-group does not arise as the derived category of perfect complexes of any algebraic variety.