Conformal groups illustrate and emphasise how rich group theory is, something usually not recognised, and they also emphasise how fundamental geometry is in physics (and conversely). Reasons, and implications for physics, are explored as a start to the study of the vast fields of mathematics and physics suggested and required. These groups have non-linear transformations, ones with singularities. Other aspects also show richness beyond what is usually realised, giving possibilities of many new insights and applications, to physics, mathematics, group theory, geometry, (non-linear) differential equations, special functions, and likely elsewhere, including optics, squeezing, quantum communication, quantum computation, cryptography, and so on. There are also suggestions about relations to the well-known elementary -- particle mass-level formula, which is reviewed. And representations of the groups for dimension greater than two have Clifford algebras as entries in their matrices. The formalism developed for quantum field theory, as a foundation for the study of conformal field theory, itself has important applications. For example while the obvious reason that the proton cannot decay -- baryon number thus lepton number must be conserved -- is almost trivial, the formalism allows a mathematical proof. Other insights provided by this and the conformal group can also be seen. How much of physics does geometry (through its transformation group) determine? As we see, a very large part. Might the geometry in which physics takes place, in which we exist, completely determine the physics within it, and conversely? These studies are motivated by that, and so attempt to raise questions, to show possibilities, to stimulate. These subjects are indeed stimulating.