Darboux transformation is a well-known technique in the study of second-order linear differential operators. Darboux transform of a second-order linear differential operator gives another such operator, which is ​"almost" isospectral to the first one –- the spectrum may be different at most at one point. We will show how to use Darboux transformation to construct new families of diffusion processes whose transition probability densities (and their spectral expansions) are given in closed form. We will also discuss connections to exceptional orthogonal polynomials. This talk is based on joint work with Minjian Yuan.