We consider the allocation of indivisible objects among agents when monetary transfers are not allowed. Agents have strict preferences over the objects (possibly about not getting any object) and are assigned at most one object. How should one allocate offices to faculty members at a university when a department moves into a new building or when the current office allocation is not considered optimal anymore? Ideally, an allocation rule would be (1) fair / equitable, (2) efficient, and (3) incentive robust. Of course, our three objectives might find different formulations depending on the exact allocation situation. Unfortunately, often the most natural properties to reflect (1) - (3) are not compatible and thus, an ideal allocation method usually does not exist..We explore trade-offs between and characterizations by various normative properties for various object allocation problems, including Shapley-Scarf exchange problems and problems where the set of objects is commonly owned by the agents.