These were (and are) educational and cultural institutions, often featuring a private subscription library and reading rooms and sometimes a lecture hall or art gallery. I suppose the proper Latin plural should be “athenaea.”
Boston Athenaeum, Boston, MA Near the top of Beacon Hill, with reading rooms looking out onto a seventeenth-century burying ground. Frequently hosts small but interesting art exhibitions. Founded 1807. Website here.
Berkshire Athenaeum, Pittsfield, MA Formerly a private athenaeum, now the local public library. The original 1876 Atheneum building, in a high-Victorian polychrome Gothic style, is now the Berkshire County Registry of Deeds. Website here.
Providence Athenaeum, Providence, RI 1836 The successor to a 1753 subscription library, it took its current name in the 19th century, at about the same time it acquired its current Greek-Revival building. Website here.
Wadsworth Atheneum, Hartford, CT 1842 Founded as primarily an art museum rather than a private subscription library, this is the only athenaeum in New England to spell its name without an “a” in its third syllable. Website here.