One of the top Italian research centre for applied physics, the Area Science Park located in Trieste, just elected its new president.
The president's CV can be found here: https://ginko.unipg.it/ws/sitopersonale/cv.php?humanid=caterina.petrillo&tipo=EV
By reading it, I discovered she got her Physics degree in 1984, starting in 1985 a post-doc bursary at an Italian institution.
Is it still possible to apply for post-doc bursaries&fundings in Italy without having completed a PhD?
The doctorate in Italy (dottorato di ricerca) was officially instituted in 1982, and the first courses started in 1983, so it was definitely not a common thing in the 1980s. People even got full professor positions without having one (since it did not exist).
Post-secondary education was not standardized across nations then like it is today; Italian scientists did their first research experiences with other research contracts, 'assistantships' (assistente) and other early-career positions that just weren't formally a PhD and did not end with discussing a thesis.
Regarding that CV, the most likely explanation is that "post-doctoral fellowship" is a liberal translation of the name of the Italian position she held. Note, though, that another non-standard practice of Italian academia is that people can legally be called dottore as soon as they get any university degree, even a 3-year bachelor. (although in the 1980s university degrees came after a single 4- to 6-year course, which was roughly equivalent to a BSc+MSc course with today's standards). So maybe in the eyes of the translator that position was "post-doctoral" because she was already a dottore (although not a dottore di ricerca).
As for your last question, it depends on the exact kind of position, but for instance I think you can still apply for a RTD-A (non-tenure-track assistant professorship) without a PhD [EDIT: not anymore since 2017; thanks @MassimoOrtolano for the correction].
About your last question, let me add something to Federico's answer. Unless you are a lawyer, and I'm not one, in Italy it is frequently difficult to properly navigate and understand the constraints imposed by the law but, as far as I understand, both the RTD-A and RTD-B positions1 require a PhD or equivalent title2. These are the current entry points if you want to aim at a tenured position.
Probably the only position that in Italy does not require a PhD is the assegno di ricerca di tipo B (the tipo A, instead, is a post-doc position), which is the typical way used to hire someone who has finished the standard three-year PhD period but hasn't yet defended the title. But if you don't have a PhD, you cannot go beyond this, and this position cannot be turned into a permanent one. Without a PhD you're also quite at risk of not being selected if someone with a PhD applies too.
1See this answer and this one for a more detailed description of these positions.
2If you open a bando (the application rules), you will likely find that the first 1-2 pages just list the relevant laws, decrees and university regulations on which the bando is based, and it's a daunting task to trace back the origin of a certain constraint.
