Conformal field theories are good friends for both physicists (since they have many applications to critical phenomena, condensed matter, and high energy physics) and mathematicians (they are mathematically well-defined). In two dimensions, conformal field theories have been studied to near-exhaustion. In three and four dimensions, much less is known, and the field is wide open. I will discuss the 'bootstrap' system of equations which, in a sense, defines conformal field theories, and recent results following from the numerical exploration of this system. I will argue that many more results lie ahead ready to be discovered by the same methods. The whole story is crying out for an analytical understanding.