Standard semantics of proper names assigns them the function of reference to individual things. This presupposes that to understand the meaning of a proper name is to understand what it denotes in the context of its referential use. But unambiguous identification of the referent of a proper name in its normal (referential) use looks like an unsolvable problem. Senses associated with referential uses of such names don't allow ascribing them singular referents in their contexts; and what concerns contexts themselves, they either don't contain proper referents at all or contain more than one, thus being referentially ambiguous. Some philosophers of language believe that the main problem for the standard semantics of proper names is that they assign them referents of wrong kind: that these, for instance, should be rather abstract entities, i.e. sets, than individual things. In this paper I try to argue that the choice of an alternative semantics for proper names does not solve the main problem with their meanings, i.e. that of their referential ambiguity. On the one hand even in the most transparent contexts there is always available an alternative referent to be assigned to the proper name's use along with some standard or default one; the fact that we prefer some standard interpretation ( when we actually do) over alternative ones reflects at best that certain semantic theories contribute more substantially to ordinary communication in comparison with others. This does not yet imply that they better explain meanings of proper names. On the other hand there is also no good explanation of a phenomenon of reference to a particular thing as such: we still don't understand how a name may become referring to one and the same particular thing in many different situations for many different agents of its use etc., and how the thing may literally become a bearer of a name.