Observers of the language have been predicting the demise of whom from about 1870 down to the present day.
Our evidence shows that no one—English or not—should expect whom to disappear momentarily; it shows every indication of persisting quite a while yet. Actual usage of who and whom—accurately described at the entries in this dictionary—does not appear to be markedly different from the usage of Shakespeare's time. But the 18th century grammarians, propounding rules and analogies, rejecting other rules and analogies, and usually justifying both with appeals to Latin or Greek, have intervened between us and Shakespeare. It seems clear that the grammarians' rules have had little effect on the traditional uses. One thing they have accomplished is to encourage hypercorrect uses of whom.
Another is that they have made some people unsure of themselves.
In current usage that refers to persons or things, which chiefly to things and rarely to subhuman entities, who chiefly to persons and sometimes to animals. The notion that that should not be used to refer to persons is without foundation; such use is entirely standard. Because that has no genitive form or construction, of which or whose must be substituted for it in contexts that call for the genitive.
Although some handbooks say otherwise, that and which are both regularly used to introduce restrictive clauses in edited prose. Which is also used to introduce nonrestrictive clauses. That was formerly used to introduce nonrestrictive clauses; such use is virtually nonexistent in present-day edited prose, though it may occasionally be found in poetry.