  This paper develops the so-called Weighted Energy-Dissipation (WED)
variational approach for the analysis of gradient flows in metric spaces. This
focuses on the minimization of the parameter-dependent global-in-time
functional of trajectories \[ \mathcal{I}_\varepsilon[u] = \int_0^{\infty}
e^{-t/\varepsilon}\left( \frac12 |u'|^2(t) + \frac1{\varepsilon}\phi(u(t))
\right) \dd t, \] featuring the weighted sum of energetic and dissipative
terms. As the parameter $\varepsilon$ is sent to~$0$, the minimizers
$u_\varepsilon$ of such functionals converge, up to subsequences, to curves of
maximal slope driven by the functional $\phi$. This delivers a new and general
variational approximation procedure, hence a new existence proof, for metric
gradient flows. In addition, it provides a novel perspective towards
relaxation.
