The phases and symmetries of a wide class of strongly-interacting chiral gauge theories are investigated. The basic constraints arise from the 't Hooft anomaly matching condition,
which is often difficult to satisfy, and as a consequence, restricts severely possible infrared behavior of the systems. Two mechanisms, the color-flavor locking and dynamical Abelianization, acting singly or in combination,
emerge from our analysis as powerful ideas which help to find a solution.