  The present work provides a mathematically rigorous account on super fiber
bundle theory, connection forms and their parallel transport, that ties
together various approaches. We begin with a detailed introduction to super
fiber bundles. We then introduce the concept of so-called relative
supermanifolds as well as bundles and connections defined in these categories.
Studying these objects turns out to be of utmost importance in order to, among
other things, model anticommuting classical fermionic fields in mathematical
physics. We then construct the parallel transport map corresponding to such
connections and compare the results with those found by other means in the
mathematical literature. Finally, applications of these methods to supergravity
will be discussed such as the Cartan geometric formulation of Poincar\'e
supergravity as well as the description of Killing vector fields and Killing
spinors of super Riemannian manifolds arising from metric reductive super
Cartan geometries.
