Generalizing semi-algebraic and global subanalytic geometry, o-minimal geometry has proven to be the natural axiomatic framework for tame geometry as envisioned by Grothendieck. Wilkie’s proofs of the o-minimality of the exponential function, as well as of restricted Pfaffian functions, are fundamental examples of o-minimal structures related to non-oscillatory solutions of differential equations. This relates in particular to Hilbert’s 16th problem: determining the number of limit cycles of planar polynomial vector fields (Roussarie, Écalle, Il’Yashenko). In the same spirit, recent works concern non-oscillatory solutions of natural functional equations, such as transexponential functions or iterated fractional exponentials (Abel, Schroeder). Some of the key tools for proving o-minimality come from resolution of singularities: preparation theorems, rectilinearization, stratification.