  We define $n$-angulated categories by modifying the axioms of triangulated
categories in a natural way. We show that Heller's parametrization of
pre-triangulations extends to pre-$n$-angulations. We obtain a large class of
examples of $n$-angulated categories by considering $(n-2)$-cluster tilting
subcategories of triangulated categories which are stable under the $(n-2)$nd
power of the suspension functor. As an application, we show how $n$-angulated
Calabi-Yau categories yield triangulated Calabi-Yau categories of higher
Calabi-Yau dimension. Finally, we sketch a link to algebraic geometry and
string theory.
