This is quite a wide timescale to have to comment on, as well as the cultural differences between different European nations. Of course there are elements of shared literary heritage between certain nations that can produce interesting points of comparison and discussion (for example, if you go far back enough in English literature studies, unless you know Old French, you'll have a hard time accessing crucial hypotexts), but I don't think 'European drama' encompasses a cohesive enough collection of texts in order to produce the kind of discussion I think you're hoping for.

I can only really discuss English literature, and then only from about 1250 at the earliest. Don't forget that, around Shakespeare's time, publication was a complicated process, and there is a great deal of debate about precisely when certain texts were written. That's why I'm reluctant to point out Marlowe, Kyd, and Jonson - all great playwrights - who were contemporary to Shakespeare, but who likely wrote plays before Shakespeare, albeit by only a few years.

Other than that, I'd like to mention 'mystery plays', the most famous of which are the 'York Mystery Plays'. These were almost like reenactments of parts of the Bible, and they would be performed by members of different guilds within the city. Performance of them coincided with the Feast of Corpus Christi. They were performed from around the middle of the 14th century, almost until Shakespeare's time.

The first answer that jumps out at me is the Roman drama. Romans had tragic and comic traditions as rich and varied as the Greeks, though they have not enjoyed the same critical longevity. Terence and Plautus wrote some great comedies, and Seneca's tragedies are worth a look. A fair amount of the Roman theatre took work from the Greeks as source material, so you'll find Oedipus and Phaedra among the tragedies.

One low comic form, Atellan farce, appears to have developed directly into the commedia dell'arte, a semi-improvisational comic form that was long popular in Italy and spread throughout Europe. Harlequinades, Pantos and Punch & Judy shows all derive from commedia. You can see some indirect influence from commedia even in some of Shakespeare.

Beyond these, the rest of the known theatrical forms are (as another poster has pointed out) liturgical dramas of various kinds and paratheatrical performances in festivals or inns. (Note: There are almost certainly some we don't know about, but for which we have no evidence.)

• Beacham, Richard C. 1996. The Roman Theatre and Its Audience. Cambridge, MA: Harvard UP. ISBN 978-0-674-77914-3.
• Brockett, Oscar G. and Franklin J. Hildy. 2003. History of the Theatre. Ninth edition, International edition. Boston: Allyn and Bacon. ISBN 0-205-41050-2.

In the medieval period, there was a "genre" of play called the "morality play" in which religious lessons were performed for an audience. The characters in the play are usually not actual people, but rather abstract concepts like Charity, Greed, etc.

The 15th century play, Everyman, is a typical one and tends to be read in a lot of English lit classes as an example of the genre. The play follows a man named Everyman (who is representative of humanity) who is told by God (via Death) that he will die soon and he needs to find someone to go with him to Heaven to plead his case. He tries to have his Beauty, his Wit, his Worldly Goods among others come with him, but all refuse/are unable to. The only "character" that is able and willing to journey with him is Good Deeds, which follows along with Catholic views on how to enter into Heaven: your good deeds, being a good person, get you in.

Morality plays tend to follow this basic outline where a character representative of humanity as a whole learns the difference between worldly constraints and heavenly ideals. They were usually performed at festivals and on feast days, often by traveling troupes. Everyman in particular isn't really "credited" to one source and there are variations of the play found throughout most of Europe, but it's one such example of "European drama".

I answered a question a while ago on the rise of Elizabethan theatre which you may find interesting, it only covers the period from the late middle ages to the early modern period and it focuses on a few countries in Western Europe - so quite a narrow scope, but it does look at the influence of Greek and Roman theatre and texts upon Renaissance theatre which seems to be relevant to your question.

http://www.reddit.com/r/AskHistorians/comments/12sy34/what_factors_lead_to_the_rise_of_elizabethan/