In this talk, we will consider critical mass criterion for two chemotaxis systems with nonlinear diffusion exponents. First, we deal with the well-known degenerate fully parabolic Keller-Segel model with in between two critical exponents and classify the global existence and finite time blow-up of weak solutions by the combination of invariant norms of initial data. Second, we provide several blow-up criteria to distinguish the global behavior of the weak solution in a degenerate two-species and cross-attraction system, specifically when it is on or below two critical curves. This talk is based on the recent study by Jose A. Carrillo and Sheng Wang.