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.

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.

The state of the world you describe is a relatively new affair. Consider for example Nordic runes, which persisted as the primary writing system in Scandinavia into the Medieval period. The dominance of Latin in the West is a consequence of how power and influence were aligned in the world in recent times; same goes for Arabic and Cyrillic.

Being incredibly old (Greek colonization of the Mediterranean and Near East happened over 2500 years ago), Greek culture got "clobbered" by subsequent waves of civilizations that overwrote or adapted that Greek influence into what worked better for them; the Cyrillic (9th century) and Arabic (6th century) scripts are much younger and the periods that drove their spread (the Russian Empire and Islamic Caliphate respectively) are younger still.

Alphabets evolve as they spread, Greek is no exception

We have a historical account of an illiterate population attempting to adopt the Greek alphabet to their language:

Being still pagans, the Slavs did not have their own letters, but read and communicated by means of tallies and sketches. After their baptism they were forced to use Roman and Greek letters in the transcription of their Slavic words but these were not suitable

— Chernorizets Hrabar, On the Letters (emphasis mine)

Despite being within the influence of the Byzantine Empire's Greek culture, the Greek alphabet (as with the Roman one) was a poor fit for the phonemes of the language. This mismatch was the very reason that Rastislav of Moravia requested of Emperor Michael III (emphasis mine):

we do not have a teacher who can explain to us in our language the true Christian faith...send us such a bishop and teacher.

The Glagolitic alphabet (and the Cyrillic that superseded it) were created specifically for Slavonic phonemes.

The same thing - to an extent - happened to Latin (itself an adaptation of Greek writing to a different set of phonemes). The Latins also did not have writing, so adapted the Greek to their purposes; in the east Persian, Aramaic, Egyptian, etc already had writing and did not need it.

Latin then continued to evolve - in much the same way that Cyrillic did - by picking up new diacritics and glyphs to accommodate the languages of cultures adopting it.

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:

• L -> Λ
• D -> Δ (in modern Greek is like "th" of course but in ancient Greece time, when adoption took place it represented the same phoneme)
• P -> Π
• S -> Σ
• F -> Φ

Also, this explains the use of approximately the same letter for different phonemes:

• X (usually as "ks" double phoneme) -> Χ (as "h" in hotel or LateX if you are familiar with that pronunciation)
• Η (as in "hotel" or no sound at all) -> Η (just another "i" sound in modern Greek or like a long εε == ee in ancient Greek)
• P -> Ρ (read as "r" in Greek)

or less impressive (they are mainly the result of a shifting in modern Greek pronunciation):

• B -> Β ("v" in modern Greek).

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:

• 1 not Latin letter: Σ -> S
• 3 letters with different pronunciation than in most languages using Latin alphabet: P read as "R" and
• Y read here (as usually is read a "I") as "F"
• H read as "I"

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:

You enjoy war

would become:

Yo enoy ar

And if we use only capital case as:

YO ENOY AR

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 adopted widely. But later was replaced with Latin or the peoples were assimiliated.

• Balkan peoples: Macedonians, Thracians, Illirians, Liburgians, Messapians, Phrygians,

• Central European people: Helvetii (mentioned by Caesar as writing with Greek alphabet) and other varieties of Celts and Germanic peoples.

• In Spain: Iberians (Ibero-Ionian script)

• In Italy: Etruscans

and a lot of other peoples wrote in their languages using Greek alphabet.

The Latin alphabet, as it has been mentioned so far, derives from a Greek alphabet.

That is, the Chalcidic alphabet. It was widely used in the areas of Chalkida and Eretria, in the island of Euboea and found its way to cities of Southern Italy through settlers of Cumae (also see this passage). From there, it went on to become the basis for Latin.

In searching for a good image, I could not find one from a widely accessed source, but here is another article by Wikipedia on archaic Greek which mentions the Chalcidic alphabet and here is an image of how the characters looked like originally.

Apart from Cyrilic which was also mentioned before, there were also significant interactions with the Coptic language.

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.