Abelian varieties are fascinating objects, combining the fields of geometry and arithmetic. While the interest in abelian varieties has long time been of purely theoretic nature, they saw their first real-world application in cryptography in the mid 1980's, and have ever since lead to broad research on the computational and the arithmetic side. The most instructive examples of abelian varieties are elliptic curves and Jacobian varieties of hyperelliptic curves, and they come naturally equipped with some additional structure, called a principal polarization. Morphisms between abelian varieties that respect both the geometric and the arithmetic structure are called isogenies. In this thesis we focus on the computation of isogenies with cyclic kernel between principally polarized abelian varieties over finite fields.