By a Schwarzian differential equation, we mean an equation of the form $S_{\frac{d}{dt}}(y) +(y')^2 R(y) =0,$ where $S_{\frac{d}{dt}}(y)$ denotes the Schwarzian derivative and $R$ is a rational function with complex coefficients. The equation naturally appears in the study of automorphic functions (such as the modular $j$-function): if $j_{\Gamma}$ is the uniformizing function of a genus zero Fuchsian group of the first kind, then $j_{\Gamma}$ is a solution of some Schwarzian equation. In this talk, we discuss recent work towards the proof of a conjecture/claim of P. Painlev\â e (1895) about the irreducibility of the Schwarzian equations. We also explain how, using the model theory of differentially closed fields, this work on irreducibility can be used to tackle questions related to the study of algebraic relations between the solutions of a Schwarzian equation. This includes, for example, obtaining the Ax-Lindemann-Weierstrass Theorem with derivatives for all Fuchsian automorphic functions.