Greco-Roman world:

The main one that has turned up in the last few years has been a set of commentaries by Origen on the Psalms, which were in a known manuscript at Munich but had not previously been assigned to him. These were discovered in 2012. This material hasn't yet been published, though a facsimile of the complete manuscript is online (I haven't read it), so it's hard to comment on its significance.

Aside from that, the main event recently would be several new Sappho fragments that turned up last year: fragments of four new poems (five complete stanzas, in one case) and further fragments that add to our knowledge of five other poems. These are mainly of interest for reconstructing the poetic techniques of a specific poet (Sappho) and the arrangement of poems in the standard Hellenistic edition of her poetry.

These poems are the subject of a substantial panel at the SCS conference in New Orleans, which began on Thursday and ends tomorrow. The relevant abstracts are available online here. (Anyone here at the conference? Did you go to the panel?)

Aside from these, new inscriptions turn up from time to time. These are usually only of minor importance (I don't know of any very significant ones that have been published in the last year or so).