Conformal Field Theory (CFT) is a framework used to describe physical systems with no intrinsic length or energy scales. CFTs have wide applicability across theoretical physics, ranging from boiling water or magnets at criticality to the low-energy dynamics of extended objects in string theory. In this talk, I will begin by describing how CFTs can be used to understand critical phenomena, and then I will discuss a couple of recent ideas that led to tremendous progress in obtaining a quantitative understanding of various corners in the space of all possible CFTs.