  Given a linear differential equation with coefficients in $\mathbb{Q}(x)$, an
important question is to know whether its full space of solutions consists of
algebraic functions, or at least if one of its specific solutions is algebraic.
After presenting motivating examples coming from various branches of
mathematics, we advertise in an elementary way a beautiful local-global
arithmetic approach to these questions, initiated by Grothendieck in the late
sixties. This approach has deep ramifications and leads to the still unsolved
Grothendieck-Katz $p$-curvature conjecture.
