I present some recent results on efforts to constraint the variation of dark energy as a function of redshift. In particular I concentrate on observational determinations of the expansion history H(z). I review a formalism that makes use of quantities similar to the Horizon-flow parameters in inflation and is general enough that can deal with multiscalar quintessence scenarios, exotic matter components, and higher order curvature corrections to General Relativity. I show how the shape of the dark energy potential can be recovered non parametrically using this formalism and I present approximations analogous to the ones relevant to slow-roll inflation. Since presently available data do not allow a non-parametric and exact reconstruction of the potential, I consider a general parametric description. This reconstruction can also be used in other approaches followed in the literature (e.g., the reconstruction of the redshift evolution of the dark energy equation of state w(z)). Using observations of passively evolving galaxies and supernova data I derive constraints on the dark energy potential shape in the redshift range 0.1<z<1.8. Our findings show that at the 1σ level the potential is consistent with being constant, although at the same level of confidence variations cannot be excluded with current data.