  In 1959, P. Hall introduced the locally finite group $\mathcal{U}$, today
known as Hall's universal group. This group is countable, universal, simple,
and any two finite isomorphic subgroups are conjugate in $\mathcal{U}$. It can
be explicitly described as a direct limit of finite symmetric groups. It is
homogeneous in the model-theoretic sense since it is the Fraisse limit of the
class of all finite groups. Since its introduction Hall's group, and several
natural generalisations, have been widely studied. In this article we use a
generalisation of Fraisse theory to construct a countable, universal, locally
finite semigroup $\mathcal{T}$, that arises as a direct limit of finite full
transformation semigroups, and has the highest possible degree of homogeneity.
We prove that it is unique up to isomorphism among semigroups satisfying these
properties. We prove an analogous result for inverse semigroups, constructing a
maximally homogeneous universal locally finite inverse semigroup $\mathcal{I}$
which is a direct limit of finite symmetric inverse semigroups (semigroups of
partial bijections). The semigroups $\mathcal{T}$ and $\mathcal{I}$ are the
natural counterparts of Hall's universal group for semigroups and inverse
semigroups, respectively. While these semigroups are not homogeneous, they
still exhibit a great deal of symmetry. We study the structural features of
these semigroups and locate several well-known homogeneous structures within
them, such as the countable generic semilattice, the countable random bipartite
graph, and Hall's group itself.
