Just three European nations have their own scripts that are neither Latin nor Cyrillic. These are Armenians, Georgians and Greeks. All three scripts are in a sense descended from the Ancient Greek cities which gave Europe much of its philosophy and art.
Greek, Georgian and Armenian letters
All Greek, Armenian and Georgian letters each represent a single sound (consonant or vowel).
Greek has capital and non-capital letters (just like Latin). However Armenian and Georgian lacks capital letters.
History of the minor European scripts
The cities of Ancient Greece perfected philosophy and science and recording it needed writing. There were many Greek scripts at the time, every region having slightly different letters. Scripts of the western Greek cities evolved into Latin script while those of the eastern cities eventually became the modern Greek script.
The empires of later centuries made the Greek script famous well beyond Greece itself. It was the prime script of Hellenic Macedonia (Alexander the Great) and the eastern Roman Empire. Eastern Roman Empire was synonymous to Eastern Christianity and so the script was used to write much of the first Bibles.
With such importance and perceived superiority other eastern nations eyed on the Greek script to use for their own unwriten languages. Many of the area’s languages had more sounds than Greek so new alphabets were devised, however the letter=phoneme idea was left intact.
Two surviving Greek-inspired alphabets are Georgian (created ~4th century AD) and Armenian (created 4th century AD). The third one is Cyrillic (created 10th century AD) which outgrew Greek by far in its importance and now has a separate article for it in this website.
Over the centuries Greek and Armenian writing reforms were limited, but Georgian had its entire old-style alphabet (asomtavruli) replaced by modern one (mkhedruli) in the 11th century.
Greek, Georgian and Armenian nations had their empires vanquished by some 15th century and fell under Turkish and Russian yoke. After they regained total independence in the 20th century their scripts have been still used solely in their original homelands.