  We study the fixed point theory of n-valued maps of a space X using the fixed
point theory of maps between X and its configuration spaces. We give some
general results to decide whether an n-valued map can be deformed to a fixed
point free n-valued map. In the case of surfaces, we provide an algebraic
criterion in terms of the braid groups of X to study this problem. If X is
either the k-dimensional ball or an even-dimensional real or complex projective
space, 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 maps
in terms of the braid groups of X. We do not currently have a complete
characterisation of the homotopy classes of split 2-valued maps of the 2-torus
that contain a fixed point free representative, but we give an infinite family
of such homotopy classes.
