I will discuss recent developments in fast orthogonal polynomial transforms. The connection problem may be subdivided into measure-perturbing problems, such as the Chebyshev–Legendre transform, and measure-preserving problems, such as for spherical harmonics. I will present algorithms that are: numerically stable, asymptotically fast, fast in practice, and broadly applicable to all classical orthogonal polynomials and many multivariate analogues of Jacobi polynomials. I have released free and open source software in C and Julia that implements the algorithms and will demonstrate it live on nonlocal reaction-diffusion equations.