The crossover transition from weak coupling at largex ξ to strong coupling at small ξ is studied in N=2 supersymmetric gauge theory with the U(N) gauge group and Nf > N (here ξ is the Fayet - Iliopoulos parameter). We find that at strong coupling a dual non-Abelian weakly coupled N=2 theory exists which describes low-energy physics at small ξ. The dual gauge group is U(Nf-N). The dual theory has Nf flavors of light dyons, to be compared with Nf quarks in the original U(N) theory. Both theories support non-Abelian semilocal strings. In each of these two regimes there are two varieties of physical excitations: elementary fields and nonperturbative composite states bound by confining strings. These varieties interchange upon transition from one regime to the other. We conjecture that the composite stringy states can be related to Seiberg's M fields.