As exemplified by o-minimality, imposing strong restrictions on the complexity of definable subsets of the affine line can lead to a rich tame geometry in all dimensions. There has been multiple attempts to replicate that phenomenon in non-archimedean geometry (C, P, V, b minimality) but they tend to either only apply to specific valued fields or require geometric input. In this talk I will present another such notion, h-minimality, which covers all known well behaved characteristic zero valued fields and has strong analytic and geometric consequences. By analogy with o-minimality, this notion requires that definable sets of the affine line are controlled by a finite number of points. Contrary to o-minimality though, one has to take special care of how this finite set is defined, leading to a whole family of notions of h-minimality. This notion has been developed in the past years by a number of authors and I will try to paint a general picture of their work and, in particular, how it compares to the archimedean picture.