Most of western languages adopt the Latin alphabet with minor variations. Arabic letters are adopted by quite a few other languages. The same is true for Eastern Europe with Cyrillic letters. Why was not the Greek alphabet adopted by others given the Greek prominence, power, and influence in the old ages? Aside from the sciences, we hardly see any use of the Greek alphabet except in Greece.
Because the territories where those languages originated were ruled by Rome, not Athens or Byzantium, for a semi-millennium. In areas where contact with Byzantium/Constantinople dominated contact with Rome, such as Russia, Ukraine, and Belarus, the Cyrillic alphabet is primary.
Here is a map showing those countries where the Cyrillic alphabet is primary.
It was adopted widely. But later was replaced with Latin or the peoples were assimiliated.
and a lot of other peoples wrote in their languages using Greek alphabet.
Ancient Greece's energies were generally oriented towards their older civilized neighbors to the east. As such, nearly all the people they traded with or conquered already had an established writing system. The colonies and trade ties they established in the uncivilized west were almost entirely usurped by Rome in the runup and aftermath of the Second Punic war.
That being said, the Latin alphabet, which as a result of that is used by nearly all Western Indo-Europeans, was ultimately derived from the Greek alphabet. Much more recently the Cyrillic Script used for most Slavic languages (and several that aren't even Indo-European) uses Greek-inspired glyphs. If you combine these two scripts, it would be fair to say that Greek-derived alphabets are in use as the primary script in close to the majority of the land area of all inhabited continents on earth.
Well, I will provide my point of view which mostly coincides with the one of @Relaxed.
Greek alphabet was adopted by many people when Greek political and spiritual influence of Greek-speaking people was at its height. That said for example we can say that two of the most commonly used alphabet derive from Greek alphabet (that does not include them nowadays in the people using Greek alphabet though) as Latin and Cyrillic alphabet.
The adoption of an alphabet, as opposed to natural process of creating a language, is mainly a political choice. People (nowadays this means countries) choose a different alphabet, or if they don't used one up to then just choose one, which is always a political choice. Polish, Czechs and partially Serbian people use Latin alphabet to express their language. Serbian use both (official Cyrillic but in all other aspect I think Latin is predominant -any more expert can correct me on this one-) alphabets. Turkish people adopted Latin in 1923 for example which signified their attitude towards the west as opposed to the east (at that time anyway).
Often writing system that are adopted from people speaking different languages, with different phonemes produce variations of such writing systems (alphabet) in order to cope with new sounds found in the new languages. For example Greek alphabet itself derives from the Phoenician one that was altered enough to fit with the phonetic system of Greeks. Polish write "Łódź" but the first letter has nothing to do with L. It just look familiar. German write "Ä" (which stands for "AE" but uses a new letter) never found in Latin before. Other alphabets use not transcribable letters as in Turkish: ş is like harsh "s" which is different than s (also existing).
Latin is another example of a modified alphabet. And of course the alphabet as we know it is just the final (up to today that is not final as the ultimate ever) step of a process of importing letters in it. Latin did not had that many letters at first and on the other hand neither did Greek alphabet. For the evolution of Latin for example a quick view is wikipedia.
This explains the differences in letters that represent approximately the same phonemes like:
Also, this explains the use of approximately the same letter for different phonemes:
or less impressive (they are mainly the result of a shifting in modern Greek pronunciation):
Another aspect that should be mentioned is that those alphabets mentioned started with the use of capital letters (now of course rarely used in common text besides primary names, begining of sentences, titles etc). So the comparison should be done to this letters not the lower case ones where the differences are more profound: A == A but a ~= α Β == Β but b ~= β E == E but e ~= ε Ζ == Ζ but z ~= ζ Η == η but h ~= η Ι == I but i ~= ι Κ == Κ but k ~= κ etc
So, the phrase it's Greek to me (although not referring to the alphabet but we can make overlook that) could be "less" Greek to me if capital letters were used instead e.g.:
αστροναύτης (meaning astronaut in Greek) would be ΑΣΤΡΟΝΑΥΤΗΣ which has:
and all other letters are familiar.
The Latin alphabet we refer today is probably the English alphabet which has 26 letters while the Latin has only 23! So, in which degree do we use Latin alphabet today? Are all alphabets deriving from Latin alphabet the same. When it is considered a new alphabet. Remember that western alphabets derive from Latin which derived from ancient Greek which derived from Phoenician alphabet. S, can we say all those alphabets use the Phonetician one? I guess not. Not too many would accept that as a correct answer. So, we don't use Phonetician, not Greek but Latin? Well, my opinion is that depends on the definition. most people would argue that we use Latin but we already miss 3 letters if we do so: J, U and W. In other words this cannot be written in (plain) Latin:
And if we use only capital case as:
which does not make much sense!
So, I guess that explains why we Greek was (or wasn't depending of the definition) adopted by other people).
It was, for the most part. While there are many other writing systems, both the Latin and Cyrillic alphabets (and, in fact, the use of an alphabet – as opposed to other type of writing systems including abjads like the Arabic or Phoenician writing system) all derive from the Greek one.
Definitional issues aside, the Greek alphabet has been extremely influential and you do in fact see Greek-inspired letters everywhere. Even cultures that have their own writing systems (e.g. the Chinese or Arab worlds) have some limited use for the latin alphabet.
Greek script was used to write Bactrian, an Iranian language spoken in what is now Afghanistan. That is about as far as you can get from Greece.