I will review why chiral fermion anomalies in any spacetime dimension are computed by evaluating an "eta-invariant" on a closed manifold in one higher dimension. The Atiyah-Patodi-Singer index theorem then implies that both local and global gauge anomalies are detected by bordism invariants, each being classified by certain abelian groups that I will identify. Mathematically, these groups are connected in a precise way, which enables one to relate local anomalies in one gauge theory to global anomalies in another. I will discuss examples of this anomaly interplay in various dimensions. In particular I will show how the SU(2) anomaly in 4d can be derived from a local anomaly by embedding SU(2) in U(2).