I’ll answer this for Greek.
Irregular verbs are really irregular, to the extent of suppletion between different persons, and all sorts of other shenanigans.
Thematic and athematic are two different classes of regular verb. The athematic class is smaller, and has more core vocabulary verbs, so we presume it to be older; it’s like the strong/weak verb distinction in Germanic.
The thematic vowel is a vowel that connects personal inflections to the tense stem of a verb in Greek. It’s alternates between an e or an o. So, to take the paradigm verb lyō:
ly-o-ɔː > lyɔː
ly-e-es > lyeːs λύεις
ly-e-e > lyeː λύει
ly-o-ōsi > lyoːsi λύουσι
Athematic verbs have a different set of inflections, with no initial vowels, and no connecting vowel. So tithɛːmi ‘I put’:
tithe-mi > tithɛːmi τίθημι
tithe-s > tithɛːs τίθης
tithe-si > tithɛːsi τίθησι
I’m not googling Slavic, sorry. 🙂 But it’d be the same thing: historical linguists have worked out that the connecting, thematic vowel is present in the thematic verbs, and absent in the athematic vowels. They are still both regular in their own way, compared to the truly irregular verbs.