The heat kernel remains the main computational tool, but a comparison to the Feynman diagram technique is also informative. A toy model of multi-variable integrals is used to illustrate all the key features of the latter, except the UV and IR divergences, from which we move on to the field theory of a self-interacting real scalar. The connection to the Gaussian approximation in the background field method is found via how the complete collection of one-loop Feynman diagrams is equivalent to the former. We then face Wilsonian renormalization head-on, starting from the Kadanoff block spin idea, and introduce it as a coarse-graining process first in real space and then in momentum or eigenvalue space. It is a process where the memory of the high-energy modes, integrated out first, is suitably retained by the surviving low-energy modes, which leads to the crucial insight that generic quantum field theories are only effective theories and the notion of universality. We close with how this idea should be applied to particle physics, where the ultimate aim resides on the microscopic side; this desire, combined with predictability, leads us to the renormalizability hypothesis.