Thirty-five years after it was postulated as a key component of the theory of weak interactions, the Higgs boson was discovered at the CERN Large Hadron Collider in 2012. Since that time, many decay modes of the Higgs boson have been observed, indicating couplings of the Higgs boson in good accord with the simplest theoretical expectations. The situation is similar for the other very massive particle of the particle physics Standard Model, the top quark, discovered in 1995. However, we should not be complacent. The most important mysteries of these particles have not been clarified. In this lecture, I will discuss the mysteries of the Higgs boson and the top quark and their connection to possible new interactions of nature, and I will explain how precision measurements of the Higgs and the top can guide us to an understanding of their more fundamental nature.