Holomorphic functions on a domain are the solutions to a set of homogeneous partial differential
equations called the Cauchy-Riemann equation, and CR functions are the solutions to the analogous
equations on the boundary. Many problems in complex analysis can be reduced to finding appropriate
solutions to the inhomogeneous versions of these equations. These solutions have been successfully
constructed when the geometry of the domain is sufficiently simple. I hope to show how these constructions
follow a pattern based on the singular integral operators of Calder\'on and Zygmund. I then plan to
discuss some more recent examples where this well-understood paradigm breaks down.