The origin of the Georgian alphabet is controversial. Whereas the Roman and Greek alphabets are the results of slow and gradual transformations of older scripts (rather than of deliberate creation), the Georgian alphabet shows up in history pretty much out of nowhere. This makes it plausible that it was invented, either by one person or several. So the obvious question is: who did it?
Historical tradition gives two conflicting answers. The first comes from a medieval Georgian chronicle called “The Lives of the Kings of Kartli.” It tells of Parnavaz, the first Kartlian king, who reigned in the third century BC. Among other exploits, the chronicle has it that Parnavaz devised the Georgian “script” (მწიგნობრობა, mtsignobroba). Some have interpreted this to mean that he developed the Georgian alphabet, but mtsignobroba can also mean “literacy” or simply “writing.” This writing could have been writing in the Georgian alphabet, but more likely it was writing in the Aramaic alphabet, which at the time was the script of the Persians. This is confirmed by archeology, which has found pre-Christian traces of the Aramaic alphabet in Georgia, but none of the Georgian. Georgian schoolchildren are taught this story.
The second traditional answer is that the Georgian alphabet was invented in the fourth century AD by an Armenian priest named Mesrop Mashtots, the same guy who invented the Armenian alphabet. This oft-repeated claim, found in a fifth-century hagiography of Mashtots, is supported by graphical similarities between several characters of the Armenian and Georgian alphabets. However, it is suspect for several reasons. For one thing, the entire passage containing the story may well have been added at a much later date, in which case it would certainly be made up. Supposing the story is part of the original history, it is not independently corroborated by any other historical source, Armenian, Georgian, or otherwise, and this is a problem when dealing with a story which the author has motive to make up (for instance, to make Mashtots and thereby Armenia seem more glorious). Besides this, the story also has it that Mashtots needed an interpreter to spread his newly-created alphabet to the Georgians, but it’s unrealistic to imagine that an alphabet with such a tight letter-to-sound correspondence as the Georgian one could have been created by someone without a masterful command of the language. All of this is to say that there’s very little reason to believe that Mashtots was the inventor of the Georgian alphabet.
[Update 7/13/15: Apparently my opinion is not clear enough for stupid readers, so I’ll make it plain: I don’t believe that Mesrop Mashtots invented the Georgian alphabet. That story is Armenian propaganda.]
Legend also has it, by the way, that Mesrop Mashtots invented an alphabet for the Caucasian Albanians (the former inhabitants of what is now Azerbaijan, who have nothing to do with modern Azeris or European Albanians). This alphabet was not known until the 20th century, when somebody found a page of it in an old Armenian book. Not surprisingly, it looks similar to both the Georgian and Armenian alphabets.
So if it wasn’t him and it wasn’t Parnavaz, then who was it? Here things become speculative. It could be that the story about Mashtots is not a lie but an exaggeration, and Mashtots was a part of or a consultant to the team of scholars that created the Georgian alphabet. The alphabet almost certainly came about in connection with the spread of Christianity in Georgia, but that doesn’t narrow things down much. Georgian Christians could have made the alphabet to bring religion to their compatriots, or it could have been made by foreign missionaries, the way Cyril and Methodius and their followers made alphabets for the Slavs. We can safely assume that whoever invented it was fully literate in at least one major foreign language, but experts can’t even agree on whether it was Greek, Aramaic, or something else. In short, very little is known.
Actually, all of this bears only on the ancient Georgian alphabet (asomtavruli), which is distinct from the modern one (mkhedruli). The question of how the modern alphabet came about is much less speculative than the question of how the ancient one came about, but it is also much less interesting. It will suffice to say that the modern alphabet ultimately derives from a cursive form of the ancient.
For some examples of Georgian alphabet, see The Georgian Alphabet: A Gallery of Specimens.