  We study the classical spaces $L_{p}$ and $\ell_{p}$ for the whole range
$0<p<\infty$ from a metric viewpoint and give a complete Lipschitz
embeddability roadmap between any two of those spaces when equipped with both
their ad-hoc distances and their snowflakings. Through connections with weaker
forms of embeddings that lead to basic (yet fundamental) open problems, we also
set the challenging goal of understanding the dissimilarities between the
well-known subspace structure and the different nonlinear geometries that
coexist inside $L_{p}$ and $\ell_{p}$.
