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This is the first of a series of papers devoted to a thorough analysis of theclass of gradient flows in a metric space $(X,\mathsf{d})$ that can becharacterized by Evolution Variational Inequalities. We present new resultsconcerning the structural properties of solutions to the $\mathrm{EVI}$formulation, such as contraction, regularity, asymptotic expansion, preciseenergy identity, stability, asymptotic behaviour and their link with thegeodesic convexity of the driving functional. Under the crucial assumption ofthe existence of an $\mathrm{EVI}$ gradient flow, we will also prove two mainresults: the equivalence with the De Giorgi variational characterization ofcurves of maximal slope and the convergence of the Minimizing Movement-JKOscheme to the $\mathrm{EVI}$ gradient flow, with an explicit and uniform errorestimate of order $1/2$ with respect to the step size, independent of anygeometric hypothesis (such as upper or lower curvature bounds) on $\mathsf{d}$.In order to avoid any compactness assumption, we will also introduce a suitablerelaxation of the Minimizing Movement algorithm obtained by the Ekelandvariational principle, and we will prove its uniform convergence as well.

Gradient flows and Evolution Variational Inequalities in metric spaces. I: structural properties