Priorities over agents are crucial primitives in assignment problems of indivisible objects without monetary transfers. Motivated by the student assignment problem to exchange programs in Japan, we introduce the so-called prioritization problem: how does one go about allocating overdemanded goods when each agent possesses one of several attributes while priority orders are established only among agents sharing the same attribute? Other applications include rationing of medical supplies, elective surgery scheduling, visa assignment and affirmative action. We show that two types of assignment protocols stand out when basic fairness and efficiency requirements are pursued in a consistent manner when randomization is used only as a last resort.