It is assumed that near-critical complex fluids, such as polymer and ionic solutions, belong to the same universality class of criticality as simple fluids. However, the range of universal critical behavior in complex fluids is usually so small that it may not be experimentally accessible. In practice, physical properties of complex fluids in the critical region often exhibit some kind of crossover behavior rather than universal asymptotic critical behavior. The character of the crossover reflects an interplay between universality caused by long-range fluctuations and a specific supramolecular structure characterized by an additional nanoscopic or mesoscopic length scale. When the correlation length of the critical fluctuations becomes comparable to this supramolecular length scale, a specific sharp and even nonmonotonic crossover from classical (van der Waals-like) behavior to universal asymptotic behavior is exhibited. In the region far away from the critical point, where the correlation length is still smaller than the characteristic length scale, one can observe classical behavior. Ultimately, in the immediate vicinity of the critical point, the correlation length becomes dominant and one should expect universal asymptotic behavior. Such a crossover has been indeed observed in polymer and ionic solutions.