  We introduce the notion of continuous orbit equivalence for partial dynamical
systems, and give an equivalent characterization in terms of
Cartan-isomorphisms for partial C*-crossed products. Both graph C*-algebras and
semigroup C*-algebras can be described as C*-algebras attached to partial
dynamical systems. As applications, for graphs, we generalize and explain a
result of Matsumoto and Matui relating orbit equivalence and
Cartan-isomorphism, and for semigroups, we strengthen several structural
results for semigroup C*-algebras concerning amenability, nuclearity as well as
simplicity of boundary quotients. We also discuss pure infiniteness for partial
transformation groupoids arising from graphs and semigroups.
