The discovery of the Higgs-like particle at the Large Hadron Collider represents a major breakthrough in our quest to unravel the mechanism of electroweak symmetry breaking. The next step in this effort is a precise experimental determination of the properties of this new state, such as its couplings and quantum numbers. These measurements, while providing new tests of the Standard Model of particle physics, will also provide novel probes of theories that go beyond the Standard Model. I will describe in detail how measurements of the signal strength parameters (ratios of measured cross sections to their predictions in the Standard Model) constrain new physics, both from an effective field theory perspective and in the context of explicit new physics models. I will emphasize the complementarity of these measurements with direct searches at colliders as well as indirect constraints such as the precision electroweak data and the stability of the electroweak vacuum. I will also present more speculative interpretations of the current trends in the Higgs data set along with the prospects for testing these scenarios.