One of the first tricks of the off-road trade has been used for a long time, probably before any one could even engage four-wheel drive because Mr. Fourwheeldrive had not invented it yet. And like many tricks that modern off-roaders use, the first guy who found out that a tire low on air could give you better traction in various types of terrain probably figured this out by mistake. There is a definite advantage to using lower air pressure on both sand and rocks. As for mud, there are two schools of thought that both work depending on the circumstances.
One idea involves tall skinny tires, which grab at the bottom of the mud to keep the Jeep moving (assuming that the mud has a bottom). Airing down will lower the ground clearance and thus defeats the purpose of having those skinnies. So to keep your axles from playing mud anchor, you should keep those narrow tires aired up for maximum clearance. The other method is to run very wide, aired-down tires, which tend to hydroplane across the top of the mud. This method requires a light vehicle with gobs of power to keep those huge mudslingers spinning. And you better be running bead locks or those huge tires could pop right off the rim, and it aint easy or fun to reset a muddy bead on a mud-filled tire. Mud is probably the easiest terrain for popping beads (with sand and rocks trailing a close second and third, respectively), but bead locks are the way to go if you are doing any type of extreme off-roading. They may lighten your wallet, but its worth your sanity and time.
As for airing down, it is at least free, and if you can guess air pressure well and bum some compressor time from one of your pals, then so is airing upthats still no excuse for not having a tire pressure gauge in your toolbox though. If you are spending a lot of time on the trail, you are probably thinking of adding some type of onboard air system. Valve core removers are nice and quick, but you wont be the first or the last to use an environmentally friendly (and free) trailside pebble or back of the valve cap to let some air out when in a pinch.
So you say, What air pressure should I run on my rig? We say that is a loaded question, an intelligent answer that totally depends on the weight of your rig, tire size, and the terrain that you plan on running. Experiment for yourself. Below are some starting points from our experiences to help you get started.