Missing on the chest: First, lack of reversal strength and speed. This is where the speed training comes in. If you have any type of explosive strength then you should never miss off your chest unless the weight is too heavy in the first place.
The second reason for missing off the chest can be due to weak starting strength after the press command. Best things for this is low pin presses with the bar just off the chest for max effort work or as a second movement for max sets of 3 or 5 reps. Make sure to pause on the pins for a second or two. Another option would be a dead stop Spoto press. Other things to help strengthen the bottom of the bench are close grip inclines, dumbbell work, and push-ups.
Missing off your chest can also be caused by weak back muscles: If you don't have a solid base to press off, you're firing from a weak foundation. The best things for Lats and upper back would be chest supported rows, pull aparts and face pulls.
Missing halfway up: This can also be fixed with more bar speed as this will allow the lifter to bust through this sticking point. This can also be caused by weak triceps. The best max effort exercises for this problem are mid-position pin presses, two board presses, shoulder saver presses and floor presses.
Bar flying off your chest and straight back into the rack: This problem can also occur because your shoulders are stronger than the triceps. You're trying to get the load off the triceps and onto where you're the strongest and that's causing the problem. On the flip side, it can also be because your shoulders aren't strong enough to keep the bar in the right path. It really depends on what the barbell does when this happens. If it handcuffs you - Triceps. If if keeps drifting back - Delts
Another technical reason this may happen: you aren't keeping your arms under the bar. This can happen if your wrists get folded back and the bar ends up being too far behind the forearm. If this happens, then the force isn't under the bar. Make sure your form is on and bring up the lockout power with specific triceps work and high board and high pin presses for max effort work.