A number of seemingly unrelated fields of mathematics is surveyed where the Schwarzian derivative \(S(f(x))\) occurs. So, for instance, in the Sturm-Liouville equation \(\varphi''(x)+u(x)\,\varphi(x)=0\), from two linearly independent solutions \(\varphi_1,\varphi_2\), the potential \(u(x)\) is reobtained as \(u=S\left(\varphi_1/\varphi_2\right)\). In the Lorentz plane with metric \(d x\,d y\) the curvature \(\rho\) of a function graph \(y=f(x)\) satisfies \(\rho'=S(f)/\sqrt{f'}\), provided \(f'(x)>0\). \(S(f)\) also arises as an invariant, telling whether two functions \(f(x)\), \(g(x)\) of one variable are related by a linear fractional transformation. For a diffeomorphism \(f:\mathbb{R}\mathbb{P}^1\to\mathbb{R}\mathbb{P}^1\) of the projective line and any point \(t\in\mathbb{R}\mathbb{P}^1\) and 3 nearby points \(t_1,t_2,t_3\), \(S(f(t))\) measures the infinitesimal behavior of the difference of the cross ratios \([t,t_1,t_2,t_3]\) and \([f(t),f(t_1),f(t_2),f(t_3)]\) in second order, i.e., the Schwarzian is the infinitesimal version of a discrete invariant under projective transformations. The behavior of \(S(f)\) under coordinate changes shows that it is a cocycle on the group of diffeomorphisms of \(\mathbb{R}\mathbb{P}^1\). This property and a relation to the Virasoro algebra are discussed. The paper ends with outlines of multi-dimensonal versions of the Schwarzian.