The front squat is one of those great bang for your buck compound exercises that can be safer than the back squat for many populations. I tend to goblet squat my non-athlete community, which is closer to the mechanics of a front squat.
But for those of you who do front squat, I wanted to share this video by Eric Cressey on the front squat grip for three reasons.
1) If you don't use the clean grip, it's okay. I've had numerous athletes come to me incapable of handling the clean grip but were forced to do so in another strength and conditioning program because that's how "athlete's" front squat. That's BS. The best way for you grip your front squat is the best way for you to do it. If you have limitations due to flexibility in the wrist or long head of the triceps, the clean grip is not going to work for you
2) Second, is to point out the progressions of what you can go through to find your grip. In the video, Eric goes through first the clean grip, then cross face grip and finally to the clean grip with straps (I've used small towels for this as well). Follow this progression, using bar weight only, and you'll find the most comfortable grip.
3) The majority of my athletes and general population use the cross face grip over the clean grip. Like stated above, there is no "right" way to grip the front squat. The only wrong way to do it is to start with the wrong grip just because someone told you this is the only way to do it. Though slightly more load may be achieved with the clean grip, it's not worth compromising the health of the wrists and shoulders.