The Euclidean algorithm is an elementary, yet fundamental, tool in the study of properties of integers. An iteration of this algorithm leads naturally to the notion of continued fractions, which have interested mathematicians for centuries due to their beautiful properties. The aim of this talk is to illustrate how a modified version of the Euclidean algorithm, due to Hirzebruch and Jung, leads naturally to certain continued fractions that crop up when studying quotients of C^2 by unitary actions of cyclic groups. The approach will be as elementary as possible and there will be plenty of examples.