Let $\phi:S \multimap S$ be a n-valued continuous multimap on some compact surface $S$. First we classify the homotopy classes of multimaps where for most of the surfaces the classification is given in terms of the braids on $n-$strings of the surface $S$. Then we give an algebraic criterion to decide which homotopy classes contains a multimap which is fixed point free. We will focus on the cases where $S$ is a closed surface of Euler characteristic $\leq 0$. Despite the fact that the algebraic criterion is quite hard, we performe some specific calculations for the case where $S$ is the torus. The concept of Nielsen number for a surface has been developed. Then I explain the status of the Wecken property for multimaps on the torus. In fact it is an open question if there is an example of a multimap which has Nielsen number zero but it can not be deformed to fixed point free. Finally a brief exposition about the case of the projective plane should be presented. Below are some of the relevant references for our purpose.