They are not flying and they are not objects.
But people saw them :)
(you can find a colection of reports of events in my site)
The events are more correctly called: UAP - Unidentified Aerial Phenomenon.
Because I've been always skeptic I studied this subject several years ago (during months),
and I concluded that two classes of UFO events can be physically explained. (edit add: I decided to offline the content because no one care to know what I found)
IMO they are just Aerial Phenomenons.
If you are so skeptic as I am you will have to study what I've explained there.
I'm expecting some comments if you deny my view point and downvote.
(the physical papers that I've consulted in relation to ice crystals and contrails... will be included in the site in the future)
quote from "Pilot Survey Results” at NARCAP.ORG (the site is offline in the moment of writing)
This paper presents the results of a confidential aircrew survey
presented to 298 currently rated and flying commercial pilots employed
by a U. S. airline.
Table 5
Results Related to Reporting One’s Sighting
Total number of pilots having seen a UAP . .. . 16 (100%)
Number who did report the sighting . . . . . . . . 4 (25 %)
Number who did not report the sighting . . . . . 12 (75 %)
If there are about 600,000 commercial pilots currently flying in the
U.S.A. and 23 percent of them saw something they couldn’t identify
this amounts to 138,000 witnesses.
colections of reports on UFOs:
COMETA report (French association) detail the results of a study by the Institute of Higher Studies for National Defense
COMETA_part1.pdf 23 pages
COMETA_part2.pdf 47 pages
UK gov. National Arquives, UFO files Files released in March 2011
NASA ASRS Database world's largest repository of voluntary, confidential safety information provided by aviation's frontline personnel, including pilots.
National Aviation Reporting Center on Anomalous Phenomena- NARCAP
Narcap: 300 cases analysed
a few selected cases:
Chicago O'Hare airport: Narcap report
Astronautics & Aeronautics July 1971 Unidentified Object in the ~South-Central U.S., July 17,1957
L'observation du vol AF 3532 1994, Coulommiers, France (more here)
Analysis of a Photograph of a High Speed Ball of Light Jul 16, 1988 - I clearly see a contrail where sunlight is reflected.
google search "Police Chase UFO over St. Atham, Cardiff", in UK 2008
The Scientists have been divorced of the UFO problem:
Society for Scientific Exploration: Report on a Survey of the Membership of the American Astronomical Society Concerning the UFO Problem: (1994) : Part 1 and Part 2
My viewpoint of the case reported by a military pilot in the page 9 of Cometa report (pag 7 of pdf), Mirage IV - UFO encounter, March 7, 1977:
Chaumont 48° 6'40.76"N , 5° 8'23.67"E
Luxeil Air Base 47°46'59.00"N ; 6°21'1.29"E
Not reported, but it is true:
No one noticed that the path of flight was in the directon of the Full Moon, South India, and the contrail was above Switzerland (cold atmosphere, no wind for sure at the time of the event). The contrail stood straight for long enough time and transported the light of Full Moon, below the horizon, to the cockpit.
Just check the images bellow, obtained with help of HPlanet - Home Planet software and Google Maps)
Physical explanation:
A contrail is made of ice crystals that in non turbulent atmosphere descend slowly exposing the bigger faces (frequently hexagonal) parallel to the ground (they go down like a large leave falling from a tree). The normal to that face oscilates around 8º max and they offer a continuous mirror for light that jump from crystal to crystal without leaving the contrail when light is injected along the longitudinal axis of the contrail. The total reflection in the faces are due to the low angle of incidence providing a 'natural low loss optical fiber'. And that was the case of this report.
The pilots usually can not see the contrails made by their own jets. This case is an exception. The contrail was there in the right direction and the moon light (full moon) made the impossible: for brief moments the pilots saw the moon. Most of the UFO lights seen by night are made by some invisible contrail in the right direction to sunlight or, the less frequent moonlight, when the sun or moon are already below the horizon.
Position of full moon over South India at the time of the event.
Local path of the flight over France and Switzerland
At left two red points (zoom it) is the path of the Mirage over France/Switzerland and a large white stripe to mark the path of the moon ligth.

Check the report of the pilot:
They noticed a brilliant light coming towards their Mirage from their
exact right and on a collision course with them, and first thought it
may be a jet fighter, but when they radioed to the ground control at
Contrex?ville for reporting it and getting it identified, they learned
that the ground radar showed nothing and that no other aircraft was
known to be in that area. The ground controller asked them to check
their oxygen, which indicates that he thought the pilots may be
hallucinating.
The light appeared bigger and bigger as it approached their aircraft
from their rear right. The pilot was flying at Mach 0.98 and made a
turn to the right and then to the left to make sure the light was not
a reflection of some sort on the cockpit. As he did these maneuvers,
both of the crew could distinguish that the light was the front of a
dark solid object. Despite the evasive maneuver, the unidentified
object managed to stay exactly behind them for a few seconds, a very
dangerous situation if the unknown object were to be hostile. The then
object made a turn to the North-West at an estimated speed of Mach II,
and went away to the left of the Mirage IV.
and reappeared 45 secs later to tease again the pilots.
In my oppinion this report of the pilots is credible. Also credible is the existence of the contrails, even if we can not see them by night time.
Incredible, IMO, is the lack of imagination of the scientific community and others, that in the absence of a theory try to put down the testimonys, or waiting for help from some alien people.
By day time the testimonys are mostly correct but the physical explanation is based in the well known (in this moment I dont have the name in my head)... effect, and ....
They report 'objects' but they saw 'light'.
Find a suitable source of Light and a path thru the medium (atmosphere).