  The paper is devoted to generalizations of actions of topological groups on
manifolds. Instead of a topological group, we consider a local topological
group generalizing the notion of a~germ or a~neighborhood in a topological
group. The notion of an action of a local group on a topological space is
introduced.
  The paper constructs the theory of local sharply $n$-transitive groups and
local $n$-pseudofields. Local sharply $n$-transitive groups are reduced to
simpler algebraic objects -- local $n$-pseudofields, similarly to the way Lie
groups are reduced to Lie algebras, and sharply two-transitive groups, are
reduced to neardomains. This can be useful, since, opposite to locally compact
and connected sharply $n$-transitive groups, which are absent for $n > 3$,
local sharply $n$-transitive groups exist for any $n$, for example, the group
$GL_n(\mathbb{R})$. Being boundedly sharply $n$-transitive, the groups under
consideration are also Lie groups, which gives extra methods for their study.
