This volume applies pre-existing techniques from singularity theory, especially unfolding theory and classification theory, to bifurcation problems. This text is the first in a two volume sequence and the focus of this book is singularity theory, with group theory playing a subordinate role. The aim is to make singularity theory more available to applied scientists as well as to mathematicians. Three case studies illustrate how singularity methods are used in applications. The authors have devoted effort to explaining the underlying mathematics in relatively simple terms and have included many examples. It is intended to be useful for a wide audience.