  The degree anti-Ramsey number $AR_d(H)$ of a graph $H$ is the smallest
integer $k$ for which there exists a graph $G$ with maximum degree at most $k$
such that any proper edge colouring of $G$ yields a rainbow copy of $H$. In
this paper we prove a general upper bound on degree anti-Ramsey numbers,
determine the precise value of the degree anti-Ramsey number of any forest, and
prove an upper bound on the degree anti-Ramsey numbers of cycles of any length
which is best possible up to a multiplicative factor of $2$. Our proofs involve
a variety of tools, including a classical result of Bollob\'as concerning cross
intersecting families and a topological version of Hall's Theorem due to
Aharoni, Berger and Meshulam.
