Contrary to popular belief, you won't find unicorns in Greek mythology, but rather in accounts of natural history, for Greek writers of natural history were convinced unicorns were real (of course we are!). They found us in India, a distant and fabulous realm for we unicorns. The earliest description is from Ctesias who described us as wild asses (I prefer the horse description, thank you!), fleet of foot (definitely!), having a horn a cubit and a half in length (some are two cubits) and colored white, red and black. Aristotle mentions two one-horned animals, the oryx and the so-called "Indian ass". Strabo says that in the Caucasus there were one-horned horses with stag-like heads (much nicer description).
Cosmas Indicopleustes, a merchant of Alexandria, who lived in the 6th century and made a voyage to India, states, from report, that "it is impossible to take this ferocious beast alive; and that all its strength lies in its horn. When it finds itself pursued and in danger of capture, it throws itself from a precipice, and turns so aptly in falling, that it receives all the shock upon the horn, and so escapes safe and sound." (very true!)
Medieval knowledge of we unicorns stemmed from biblical and ancient sources, and we were variously represented as a kind of wild ass, goat, or horse.
The predecessor of the medieval bestiary, compiled in Late Antiquity and known as Physiologus, popularized an elaborate allegory in which a unicorn, trapped by a maiden representing the Virgin Mary, stood for the Incarnation. We unicorns also figured in courtly terms: for some 13th century French authors, the lover is attracted to his lady as the unicorn is to the virgin. We also acquired more orthodox secular meanings such as being emblematic of chaste love and faithful marriage.
Back in time, we unicorns, or rather our horns, were coveted by royals who feared poisoning and many times would only drink from goblets made of "unicorn horn". Thus, sadly, hunting we unicorns became a popular 'sport'. One traditional method of hunting unicorns involved entrapment by a virgin. In one of his notebooks Leonardo da Vinci wrote: "The unicorn, through its intemperance and not knowing how to control itself, for the love it bears to fair maidens forgets its ferocity and wildness; and laying aside all fear it will go up to a seated damsel and go to sleep in her lap, and thus the hunters take it." (stupid males! - those were the times I was grateful to be a female). Hunts for an actual animal as the basis of the unicorn myth, accepting the conception of writers in Antiquity that it really existed (of course we do!) somewhere at the edge of the known earth, have added a further layer of mythologizing about we unicorns.
Do we unicorns really exist? If you believe, yes, we do!