Originally Posted by Wilfred Laurier

It's possible on the front but you will likely have to get a new fork. Depending on the bike it may be easy or very difficult to find a fork that fits.

It is possible on the rear, but only if you are willing to get disc brake mounts welded or brazed onto your frame.

These possibilities are based on the assumption that you don't already have disc brake mounts.

Generally speaking, caliper brakes that come stock on most road-oriented bikes are more than strong enough on the rear - you can lock up the rear wheel to skid, and no braking power beyond that is useful. I believe a good disc brake on the front adds a bit of stopping power and control, but most people using modern dual-pivot caliper brakes find them sufficient on the front as well.

If your bike was older or very inexpensive, you may have the older style single-pivot caliper brakes, or the ancient style centre-pull caliper brakes.. Upgrading to decent dual pivots would give you much better braking and would be a bolt-on upgrade requiring no modifications.