Many C*-algebras of interest are generated by a set of partial isometries. Often, one can arrange for such a generating set to be closed under multiplication and adjoint - such a set is necessarily an inverse semigroup. Even more, such examples frequently turn out to be universal for some class of representations of its generating inverse semigroup. We will discuss the universal C*-algebra generated by an inverse semigroup, and its natural boundary quotient as defined by Exel. If time permits, we will discuss how one can often phrase properties such as simplicity and pure infiniteness for the quotient in terms of algebraic properties of the inverse semigroup; this approach recently allowed for simplicity criteria to be given for certain Cuntz-Li algebras.