  We consider relativistic and non-relativistic particles and strings in spaces
(or space-times) with a degenerate metric. We show that the resulting dynamics
is described by a rich structure of constraints. All constraints are classified
and the dynamics depends strongly on the parity of the difference between the
dimension of the space (or space-time) and the rank of the degenerate metric.
For a particular class of degenerate metrics we can identify the null
eigenvectors of the metric with its Killing vectors. We also give the first
steps towards the quantization of the non-relativistic particle using the
Senjanovic path integral quantization and the Batalin-Fradkin-Tyutin conversion
method.
