Dragon can refer to many different types of creature - for instance, the Komodo Dragon is a carnivorous lizard that can grow three meters long, and does exist. Paleontology gives us the Pterodactyl, an extinct flying lizard with a wingspan of over 10 meters, as well as the Tyrannosaur, an extinct seven ton killing machine. Either of those could fit some definitions of 'Dragon', and did exist.
Most dragons, however, are portrayed in incredibly fantastic ways - gigantic flying creatures with human intelligence that can breathe fire. This is the product of myth and legend, and no evidence of such a creature actually living has been found.
There are many world cultures that also believe in the existence of dragons as spirits (many from Eastern Asia; China and Japan most notably), and that they do not always need to take a physical form. So, the answer may also change based on your personal spiritual beliefs.