Paleo-European languages
Old European is a designation for the (mostly unknown) languages that were spoken in Europe prior to the spread of the Indo-European family which dominates the continent today. In this sense, Basque and the Caucasian languages are Old European languages (however, the languages spoken today are certainly not identical with the languages that were spoken before the spread of Indo-European - they certainly changed a lot over time). The term Old European, however, is often used more narrowly with reference to the unknown languages of the first Neolithic farmers in central Europe, who appear to have immigrated from the east around the year 5500 BC. Their original homeland probably no longer exists: it is drowned beneath the Black Sea, and was where now is the Bay of Odessa. (Before about 5500 BC, the Black Sea was a freshwater lake - the Euxine Lake - with a level much lower than the present day sea level. Then the rising sea burst through the Bosporus and flooded the Black Sea basin within a few years to almost the present day level.)
These Old European languages are not attested in writing (but see Old European script for a set of undeciphered signs that were used in the easternmost parts of the Old European area, which may or may not have been a writing system, but are at any rate undeciphered). The only access to it we have are place names and especially river names that are found all over central and western Europe, and possibly loanwords in the Indo-European languages now spoken there. The area across which these Old European river names occur largely coincides with the area where remnants of the central European Neolithic (the so-called Linear Pottery culture) and its daughter cultures can be found. There is an especially good correlation between the areas of the Old European hydronymy and the Beaker culture.
Hypotheses about the relationships of Old European
There are several hypotheses concerning the relationships of Old European to other, attested languages.
The German linguist Hans Krahe ascribed the Old European hydronymy to an Indo-European language that, according to him, was the common ancestor of Italic, Celtic, Germanic, Baltic and possibly Slavic.
However, there are two problems with this hypothesis. First, these languages are not characterized by any shared innovation and do not seem to form a valid node in the Indo-European family tree; second, the Old European river names, while seemingly having Indo-European etymologies, do not reflect the characteristic sound changes of the languages in which they are found and are thus more likely to be borrowed rather than inherited. For example, the Old European river names show a predominant /a/-vocalism which looks quite un-Indo-European, while the vowels /e/ and /o/ which are frequent in Indo-European appear to have been absent.
Another German linguist, Wolfgang Paul Schmid, addressed the first problem by assuming that Krahe's Old European was Proto-Indo-European itself, which he localized in central Europe. This is widely rejected today, and it doesn't solve the second problem.
The second problem was addressed by Theo Vennemann (again, a German linguist) who proposed that Old European was not an Indo-European language at all, but related to Basque. According to Vennemann, 'Vasconic' languages were spoken all across continental Europe west of a line that approximately runs from Riga to northern Dalmatia. (East of that line, Uralic was spoken in the north and Indo-European in the south; in the British Isles, Vennemann assumes, an Afro-Asiatic language was spoken.) However, Vennemann's Vasconic etymologies are very weak, and the distribution of the Old European river names shows a gap in Gascony, where Basque was spoken in antiquity. Another problem is that the Old European hydronymy is also well-attested in the British Isles, where Vennemann assumes an entirely different language (of the Afro-Asiatic family) to have been spoken. (This conflict, however, is not entirely irreconcilable if one assumes two subsequent substrata.)
Jörg Rhiemeier assumes that the language in question, Hesperic, was related to, but distinct from, Indo-European. The /a/-centred vocalism of the unknown language preserves a state that existed in Proto-Indo-European before the rise of ablaut. Together, Hesperic and Indo-European form the Europic language family.