The dynamic behaviors of the solutions for the reverse-space-time derivative nonlinear Schrödinger equation are studied by Darboux transformation. The breathers on the periodic and double-periodic background are derived by the N -fold Darboux transformation. The rogue waves on the periodic and double-periodic background are constructed by the generalized Darboux transformation. It is worth mentioning that the breathers and rogue waves on double-periodic background based on the plane wave seed solution are first constructed. The two peak, four peak rogue waves on the double-periodic background are found. And the rogue waves on the double-periodic background can be transformed into the classical rogue wave on the plane wave background with a special reduction relation.