These animals show up in art that dates back more than 5,500 years. The first Griffin-type creatures appear in Persian and Egyptian art over five millennia ago, about by about 1,500BCE full Griffins as we know them today were popping up in Greece and Central Asia. For the past several thousand years they have been featured in various works of art, including sculpture and architecture, and they have shown up in numerous writings as well (including those of Pliny the Elder).
In most mythologies, Griffins are wise and powerful creatures that command a great deal of respect. They are often said to seek out and guard various treasures, and their strength and wisdom has led them to become popular figures in heraldry. Don't confuse them with Hippgriffs though! Those are very different animals, with the bodies of horses rather than lions. Hippogriffs are said to me the crosses of Griffins and Mares.
There is actually a pretty decent theory surrounding the origin of the Griffin myth. Folklorist Adrianne Mayor has speculated that fossils of Protoceratops confused the ancient people, leading to the invention of this mythical beast.