  In this paper we propose a general functorial definition of the operation of
\emph{local tropicalization} in commutative algebra. Let $R$ be a commutative
ring, $\Gamma$ a finitely generated subsemigroup of a lattice, $\gamma : \Gamma
\rightarrow R/ R^*$ a morphism of semigroups, and $\V(R)$ the topological space
of valuations on $R$ taking values in $\R \cup \infty$. Then we may
\emph{tropicalize} with respect to $\gamma$ any subset $\W$ of the space of
valuations $\V(R)$. By definition, we get a subset of a rational polyhedral
cone canonically associated to $\Gamma$, enriched with strata at infinity. In
particular, when $R$ is a local ring, $\gamma$ is a \emph{local} morphism of
semigroups, and $\W$ is the space of valuations which are either positive or
non-negative on $R$, we call these processes \emph{local tropicalizations}.
They depend only on the ambient toroidal structure, which in turn allows to
define tropicalizations of subvarieties of toroidal embeddings. We prove that
with suitable hypothesis, these local tropicalizations are the supports of
finite rational polyhedral fans enriched with strata at infinity and we compare
the global and local tropicalizations of a subvariety of a toric variety.
