Can you challenge a referee's call in fencing, and how does video replay work?
You cannot argue a judgment call, but where video replay is set up you can formally ask the referee to look at the touch again. Everything about what happened on the strip, such as who had right of way, whether a hit was valid or whether someone stepped off the side, counts as a point of fact. The rules say a referee's decision on a point of fact cannot be appealed except through video review, and openly casting doubt on that kind of call is itself an offense that earns a yellow card.
Video replay only exists where the organizer has installed it. Under the international rules it is compulsory at world cups, grand prix events, world championships and the Olympics, usually from the later rounds, and USA Fencing applies the same replay rules only at events where video is in use. Most local tournaments have none. Where it exists, only the fencer on the strip can ask, never the coach. In an individual event you get one request per pool bout and two per direct elimination bout, and in a team match one per relay. If the referee changes the call, you keep your request. If the call stands, you lose it.
When you ask, the referee walks over to the video referee and they watch the action together, up to four replays at any speed, including slow motion. The referee then gives the final decision, and nobody can ask for that same action to be reviewed again. The referee can also check the monitor on their own, the video referee can suggest a look, and when the score is tied for the deciding touch, the referee must check the video before calling it. At the biggest finals the replays are shown on a screen for the crowd.
Without video, your only formal challenge is about the rules themselves, not about what the referee saw. If you believe the referee misunderstood or misapplied a definite rule, say so courteously and right away, before the next touch is fenced. If the referee keeps their view, the head referee settles it, and an appeal judged unjustified is penalized. The shouting and fist pumps you see on TV are not appeals, just emotion and showmanship, so save your formal requests for touches you truly believe were called wrong.