We study the effect of polydispersity on the structure of polymer brushes by analytical theory, a numerical self consistent field approach, and Monte Carlo simulations. The polydispersity is represented by the Schulz-Zimrn chain-length distribution. We specifically focus on three different polydispersities representing sharp, moderate, and extremely wide chain length distributions and derive explicit analytical expressions for the chain end distributions in these brushes. The results are in very good agreement with numerical data obtained with self-consistent field calculations and Monte Carlo simulations. With increasing polydispersity, the brush density profile changes from convex to concave, and for given average chain length N-n and grafting density sigma, the brush height H is found to scale as (H/H-mono - 1) alpha (N-w/N-n - 1)(1/2) over a wide range of polydispersity indices N-w/N-n (here H-mono is the height of the corresponding monodisperse brush). Chain end fluctuations are found to be strongly suppressed already at very small polydispersity. On the basis of this observation, we introduce the concept of the brush as a near-critical system with two parameters (scaling variables), (N-n sigma(2/3))(-1) and (N-w/N-n - 1)(1/2), controlling the distance from the critical point. This approach provides a good description of the simulation data. Finally, we study the hydrodynamic penetration length l(p) for brush-coated surfaces in flow. We find that it generally increases with polydispersity. The scaling behavior crosses over from l(p) similar to N-n(1/2)sigma(-1/6) for monodisperse and weakly polydisperse brushes to l(p) similar to N-n(2/3) for strongly polydisperse brushes.