  We present an application-oriented approach to Urysohn and Hammerstein
integral operators acting between spaces of H"older continuous functions over
compact metric spaces. These nonlinear mappings are formulated by means of an
abstract measure theoretical integral involving a finite measure. This flexible
setting creates a common framework to tackle both such operators based on the
Lebesgue integral like frequently met in applications, as well as e.g.\ their
spatial discretization using stable quadrature/cubature rules (Nystr"om
methods). Under suitable Carath{\'e}odory conditions on the kernel functions,
properties like well-definedness, boundedness, (complete) continuity and
continuous differentiability are established. Furthermore, the special case of
Hammerstein operators is understood as composition of Fredholm and Nemytskii
operators. While our differentiability results for Urysohn operators appear to
be new, the section on Nemytskii operators has a survey character. Finally, an
appendix provides a rather comprehensive account summarizing the required
preliminaries for H\"older continuous functions defined on metric spaces.
