  In this article, we focus on the inverse scattering transformation for the
Fokas-Lenells (FL) equation with nonzero boundary conditions via the
Riemann-Hilbert (RH) approach. Based on the Lax pair of the FL equation, the
analyticity and symmetry, asymptotic behavior of Jost solutions and scattering
matrix are discussed in detail. With these results, we further present a
generalized RH problem, from which a reconstruction formula between the
solution of the FL equation and the Riemann-Hilbert problem is obtained. The
N-soliton solutions of the FL equation is obtained via solving the RH problem.
