  Although there are very algorithms for embedding graphs on unbounded grids,
only few results on embedding or drawing graphs on restricted grids has been
published. In this work, we consider the problem of embedding paths and cycles
on grid graphs. We give the necessary and sufficient conditions for the
existence of cycles of given length $k$ and paths of given length $k$ between
two given vertices in $n$-vertex rectangular grid graphs and introduce two
algorithms with running times O$(k)$ and O$(k^2)$ for finding respectively such
cycles and paths. Also, we extend our results to $m\times n\times o$ 3D grids.
Our method for finding cycle of length $k$ in rectangular grid graphs also
introduces a linear-time algorithm for finding cycles of a given length $k$ in
hamiltonian solid grid graphs.
