A systematic method is developed to obtain localized wave solutions of the Fokas–Lenells equation on theta-function backgrounds. First, using the properties of the theta-functions, we find a theta-function seed solution of the Fokas–Lenells equation. Next, based on the Riccati equation of Lax pair, we construct a Darboux transformation of the Fokas–Lenells equation. Then, the Kaup–Newell type spectral problem with theta-function potentials is solved by using the Baker–Akhiezer functions in the algebro-geometric method. Finally, Exact rogue-wave and breather solutions of the Fokas–Lenells equation on theta-function backgrounds are constructed by using the derived Darboux transformation and Baker–Akhiezer functions. In addition, the interaction dynamics of various localized wave solutions are analyzed by choosing appropriate parameters.