As someone who has seen a full sized saucer craft rising straight up in the daylit sky near me for several minutes, I can vouch that they probably do exist. Flying saucers, that is.
There is the possibility that my eyes, along with the people's who were next to me, were mistaken. I don't know what, if anything, was inside operating it, but flying saucers are often associated with aliens and the government. Maybe aliens exist, but my feeling is that what I saw was government involved.
I was waiting to see if someone would mention a personal experience before mentioning mine. I've also had a similar experience when I was in my early twenties, which was before cellphones and smartphones. The event was witnessed by dozens of other people around me and by other people in the surrounding area. It was 1993 and it happened in a populated city in Texas that was nationally known for unusual UFO sightings.