  We study the existence and properties of rogue wave solutions in different
nonlinear wave evolution models that are commonly used in optics and
hydrodynamics. In particular, we consider Fokas-Lenells equation, the
defocusing vector nolinear Schr\"odinger equation, and the long-wave-short-wave
resonance equation. We show that rogue wave solutions in all of these models
exist in the subset of parameters where modulation instability is present, if
and only if the unstable sideband spectrum also contains cw or zero-frequency
perturbations as a limiting case (baseband instability). We numerically confirm
that rogue waves may only be excited from a weakly perturbed cw whenever the
baseband instability is present. Conversely, modulation instability leads to
nonlinear periodic oscillations.
