The British Open Championship cut rule reduces the 156-player field down significantly heading into the weekend.
The British Open Championship cut line is determined in a way similar to what you'll see week to week on the PGA Tour, with the top 70 players and ties getting into the final two rounds. There is no 10-shot rule, which means there's no guarantee that a player within 10 shots of the lead qualifies for the weekend rounds. There is a 10-shot rule at the Masters.
There's also no secondary cut if 78 or more professionals make the cut, as would be the case in a standard PGA Tour event.
From 1968 through 1985, the British Open made two cuts, first to the low 80 and ties after 36 holes, then down to the low 60 and ties after the third round.