In a recent paper arXiv:1602.02300v2, Cook II, Harbourne, Migliore and Nagel related the splitting type of a line arrangement in the projective plane to the number of conditions imposed by a general fat point of multiplicity $j$ to the linear system of curves of degree $j+1$ passing through the configuration of points dual to the given arrangement. If the number of conditions is less than the expected, we say that the configuration of points admits unexpected curves. In this paper, we characterize supersolvable line arrangements whose dual configuration admits unexpected curves and we provide other infinite families of line arrangements with this property.