We study the dynamics of the group of isometries of $L_p$-spaces. In particular, we study the canonical actions of these groups on the space of $delta$-isometric embeddings of finite dimensional subspaces of $L_p(0,1)$ into itself, and we show that for $p eq 4,6,8,ldots$ they are $varepsilon$-transitive provided that $delta$ is small enough. We achieve this by extending the classical equimeasurability principle of Plotkin and Rudin. We define the central notion of a Fraisse Banach space which underlies these results and of which the known separable examples are the spaces $L_p(0,1)$, $p eq 4,6,8,ldots$ and the Gurarij space. We also give a proof of the Ramsey property of the classes ${ell_p^n}_n$, $p eq 2,infty$, viewing it as a multidimensional Borsuk-Ulam statement. We relate this to an arithmetic version of the Dual Ramsey Theorem of Graham and Rothschild as well as to the notion of a spreading vector of Matouv{s}ek and R{o}dl. Finally, we give a version of the Kechris-Pestov-Todorcevic correspondence that links the dynamics of the group of isometries of an approximately ultrahomogeneous space $X$ with a Ramsey property of the collection of finite dimensional subspaces of $X$.
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