In mathematized physics quantities referred to as "the constants of nature" or "physical constants" have long played an important role, e.g. the gravitational constant, the speed of light, and the charge of an electron. It is surprising, then, that philosophers of science have given little serious thought as to their nature. In this talk, I will give an initial characterization of these quantities and will present some of the open questions and choice points regarding their nature(s). I will present genealogical, formal, and nomological accounts of the constants of nature. Most of the recent controversy regarding the constants concerns this final aspect: What is the relation between the constants and the laws of nature? I will show that an examination of this question leads to a reconsideration of some fundamental aspects of the modality of physical law. If all goes well, a new and fundamental problem space should be opened for all of us to explore.