As one of the most popular reliability models, the previous several
decades have witnessed remarkable developments and extensive applications of
consecutive k systems, and a number of related models have been developed. In
the paper, a summary of the state of the arts in the field is provided. After a brief
introduction of conventional consecutive k systems, we focus on variants of the
consecutive k systems by considering failure criteria (single failure criterion and
multiple failure criteria), geometric structure of the system, states of components
and the system, weight of each component, dependency of components. Finally,
several future challenges deserving further studies are highlighted.