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We study the fixed point theory of n-valued maps of a space X using the fixedpoint theory of maps between X and its configuration spaces. We give somegeneral results to decide whether an n-valued map can be deformed to a fixedpoint free n-valued map. In the case of surfaces, we provide an algebraiccriterion in terms of the braid groups of X to study this problem. If X iseither the k-dimensional ball or an even-dimensional real or complex projectivespace, we show that the fixed point property holds for n-valued maps for all n$\ge$ 1, and we prove the same result for even-dimensional spheres for all n$\ge$ 2. If X is the 2-torus, we classify the homotopy classes of 2-valued mapsin terms of the braid groups of X. We do not currently have a completecharacterisation of the homotopy classes of split 2-valued maps of the 2-torusthat contain a fixed point free representative, but we give an infinite familyof such homotopy classes.

Fixed points of n-valued maps, the fixed point property and the case of surfaces -- a braid approach