Abstract : In this paper, we explore the fixed point theory of n-valued maps using configuration spaces and braid groups, focussing on two fundamental problems, the Wecken property , and the computation of the Nielsen number. We show that the projective plane (resp. the 2-sphere S^2) has the Wecken property for n-valued maps for all n ∈ N (resp. all n ≥ 3). In the case n = 2 and S^2 , we prove a partial result about the Wecken property. We then describe the Nielsen number of a non-split n-valued map φ : X ⊸ X of an orientable, compact manifold without boundary in terms of the Nielsen coincidence numbers of a certain finite covering q : X^ −→ X with a subset of the coordinate maps of a lift of the n-valued split map φ • q : X^ ⊸ X. In the final part of the paper, we classify the homotopy classes of non-split 2-valued maps of the 2-torus, and we give explicit formulae for the Nielsen number of such 2-valued maps using our description of it.