I am pretty anti conspiracy theories mostly. The Jill Dando theory is plausible but hardly a conspiracy theory as her murderer is still at large.
I do think that Lee Harvey Oswald shot Kennedy but there is evidence that someone else was involved. Look it up on Wikipedia to get the proper details but:
JFK and Jackie were in the back seat of the car. In front of them were Governor Connolly and his wife. There were supposedly three shots fired by Oswald. One missed, one hit Kennedy in the back, exited through his throat and hit Connolly. The third was the fatal shot. HOWEVER both Connolly and his wife were adamant that the shot that hit him [Connolly] was not the shot that hit JFK through the back/throat. They both remember hearing that shot, turning round to see JFK clutching his throat, and then Connolly was shot. So, hey , four shots not three! So what? BUT the cine film that was made of the assassination shows the timescale. It shows the car going behind a sign and disappearing from view with all four passengers fine. When it reappears both Connolly and JFK have been shot. The problem is that the rifle Oswald used could not have fired twice in the amount of time the car is lost from view in the film. This means that either:
a) there was more than one gunman
b) the same bullet hit JFK and Connolly. But Connolly AND his wife say that was not the case.
When the Warren commission came out it contained the single bullet theory. Gov and Mrs Connolly accepted the report but maintained their version of events. 
My humble take on this was that Oswald was a gunman on the day and may indeed have fired the fatal shot. However I think there may well have been another gunman and the reason I think it was hushed up was because Oswald, despite being a communist was at least an American. If the other gunman was a Russian it could have led to World War 3. Far better to blame it all on an American loser than destroy the whole world. A noble act rather than a conspiracy.
FWIW I am not sure the Russians would have been responsible anyway. They would have been watching JFK closely and his sexual behaviour was so appalling it would have been very easy to discredit him, far easier than killing him.