Abstract: This talk is about a class of two-dimensional stochastic differential equations (SDEs) of McKean--Vlasov type in which the conditional distribution of the second component given the first appears in the equation for the first component. Such SDEs arise when one tries to invert the Markovian projection developed by Gyöngy (1986) and Brunick-Shreve (2013), typically to produce an Itô process with the same fixed-time marginal distributions as a given one-dimensional diffusion but with richer dynamical features. Variants of the SDEs discussed in this paper enjoy frequent application in the calibration of local stochastic volatility models in finance, despite the very limited theoretical understanding. This talk will give an introduction to this intriguing topic in which many open questions remain. In particular, I will highlight some recent results, obtained jointly with Mykhaylo Shkolnikov and Jiacheng Zhang, on the strong existence of stationary solutions for these SDEs, as well as their strong uniqueness in an important special case.