  We define and study embeddings of cycles in finite affine and projective
planes. We show that for all $k$, $3\le k\le q^2$, a $k$-cycle can be embedded
in any affine plane of order $q$. We also prove a similar result for finite
projective planes: for all $k$, $3\le k\le q^2+q+1$, a $k$-cycle can be
embedded in any projective plane of order $q$.
