**Abstract:** This lecture mainly concerns the question as to whether the property of a separable Banach space to be reflexive can be characterized metrically. Broadly speaking, a metric characterization of a property of Banach spaces is an equivalent formulation of this property that refers only to the metric structure of the space and not to its linear structure. On each Schreier family $\mathcal{S}_\alpha$, $\alpha<\omega_1$ we define two metrics $d_{\infty,\alpha}$ and $d_{1,\alpha}$ and we study the separable Banach spaces $X$ for which there exists a map $\Phi:\mathcal{S}_\alpha\to X$ and two positive constants $c$, $C$ so that for all $A,B\in\mathcal{S}_\alpha$ \begin{equation} cd_{\infty,\alpha}(A,B)\leqslant \|\Phi(A)-\Phi(B)\|\leqslant C d_{1,\alpha}(A,B). \end{equation} As it turns out, such maps can always be constructed on non-reflexive Banach spaces. However, within the class of separable reflexive Banach spaces the existence of a map with the above property is closely linked to the Szlenk index of that space, an index measuring the ``size'' of the space's dual. We draw two main conclusions. The first one concerns a metric interpretation of reflexivity, namely the following: a separable Banach space is reflexive if and only if for every $\alpha<\omega_1$ there exists a map satisfying the above property. The second one concerns a metric characterization of the Szlenk index of a reflexive Banach space. More precisely, for countable ordinal numbers $\alpha$ with the property $\alpha = \omega^\alpha$ it follows that for a separable reflexive Banach space $X$, $(\mathcal{S}_\alpha,d_{1,\alpha})$ bi-Lipschitzly embeds into $X$ if and only if $\max\{\mathrm{Sz}(X),\mathrm{Sz}(X^*)\}>\alpha$. This is joint work with Thomas Schlumprecht.