First, try everything possible to not break the bolts off. Soak it (for a long time), rap it, heat it, repeat. Depending on the size of the bolt you'll get a "feel" for how much you can crank it before it will break off - stop before you get there.
If you've already broken it (now you're a hammer mechanic!): As a maintenance machinist I've seen it all. If any of the stud is still sticking out it's a good idea to take a file and make the top flat (easier to center punch accurately). You need to get a fairly sharp center punch dead center. Start with a light tap and look at it. If it's off you can move the dimple around by putting the center punch at an angle and hitting a little harder. Don't start drilling until you know it's dead center!
Next, start with a small drill and only drill a tiny bit out. Stop and check. If it looks off you can correct the hole's direction by drilling at an angle, early on, if you see you're not going right down the center this is the only time to fix that. Once you have a small hole that's centered all the way through you can start gradually using larger and larger drills. With the small drill you want to go slowly, pull the bit out to clear out the chips, maybe a little oil to help keep things cool and help in cutting. You do not want to break off a drill in the hole so go easy!
At some point you might try a screw extractor (if you can get a large enough hole to fit the proper sized extractor with still enough meat on the side walls of the bolt). If the bolt is small a small screw extractor is very likely to break inside the hole - you don't want to do that so stop before you do! If the wall thickness is too thin the extractor will put pressure on the outside wall and you won't get the bolt out. Don't expect the extractor to work miracles but every once in a while it does work but consider yourself lucky if it does.
Often you'll just need to drill out the bolt almost to the edge of the inside threads of the hole. Once you are close you can use a few small sharp tools "picks" to unwind the threads - it will come out like a little spring. Once most of the bolt is gone you can chase the threads with a tap to clean it out.
When mechanics brought me a bolt with a drill/tap/EZ-out broken off inside their off-center hole there was no cutting it out (they are hardened steel). We had something we called a "tap burner". It was like a hollow welding rod with fluid running thru it and the arc eroded away the offending hardened miscreant. If you find yourself in that situation you should look for an automotive machine shop that has such a setup.
One more thing. On my old 7.5 hp Elgin outboard the holes for the bolts were drilled thru and threaded. Once I got most of the broken bolts out I was able to drill a thru hole and then use a bolt and nut to hold the piece together. Not pretty, but it works just as good! On my newer Evinrude I learned to be patient, use heat, impacts and soak. I didn't have any broken bolts to remove. Both engines were in a salt water environment for many years - finesse and patience, with just a little bit of technique is the key.
