Rogue waves, solitons, and breathers are different types of nonlinear waves of physical significance, which are the central objects of nonlinear science. These nonlinear modes are associated with the modulation instability (MI). The latter is a nonlinear phenomenon of fundamental importance which has attracted particular attention in many physical contexts. In this talk, I will present some results on this subject analytically and numerically. This includes 1) rogue wave generation on a localized background beam, state transitions between rogue waves and solitons, and vector breather collisions; 2) multipeak solitons on a nonvanishing background; 3) the nonlinear MI nature of superregular breathers. In particular, the exact link between the exact rogue wave solutions and the MI shows that the transition between rogue waves and solitons appears as a result of the attenuation of the MI growth rate. Moreover, we demonstrate the MI nature of superregular breathers by showing that the absolute difference of group velocities of superregular breathers coincides with the linear MI growth rate. These results could provide insight into many nonlinear mode excitations in related nonlinear systems.