The notion of effective action is introduced; the rest of the volume requires the concept crucially. The Wick rotation of the (effective) action is then given a new treatment in terms of how the metric is analytically continued. Spinors require a special treatment due to how the underlying Clifford algebra changes its character drastically in different signatures. Recasting the fermion action in Euclidean signature, we explore the Dirac operator and the eigenvalue problem thereof. The path integral of fermions is expressed in terms of the determinant or the Pfaffian of the Dirac operator, for which eigenvalue doubling and quadrupling in various dimensions are carefully studied. The heat kernel expansion shall be the main computational tool for the rest of the volume; a pedagogical overview is given with explicit formulae that suffice for all of the one-loop computations performed in this volume. The final section gives explicit comparisons of the path integral versus the Hamiltonian partition functions for bosonic and fermionic harmonic oscillators, in part as a way to demonstrate the equivalence and in part as a precursor for the index theorem that will enter later discussion.