The authors investigate the nature of bifurcations that occur in a finite dimensional smooth dynamical system depending on more than one parameter. In addition to the saddle point node which is analogous to the Hopf bifurcation, a cusp node can occur. This cusp singularity occurs only in the presence of more than one parameter. The authors obtain algebraic conditions for the family of vector fields to be versal, which enables them to predict the topologically distinct phase trajectories in the vicinity of the singularity. These results are illustrated by a number of applications to population and oscillator models.