So the bolt finally broke

Despite one's best efforts the occasional fastener is going to break. How to remove it depends on what remnants are left and why it broke. The main problem is when the remains are under the surface of the component the bolt is in and you can't get a pair of vice grips on the stub. If a bolt or stud snapped while it was being tightened then chances are the remains will not be tight in the female thread it's left in. In those cases you can drill a small hole into it and use an Easy-Out which is a left hand threaded removal tool. The tricky bit is drilling the pilot hole without slipping off and ruining the component the bolt is stuck in. That's much more easily done on a milling machine or pillar drill with a milling cutter rather than a drill bit which will slip off the jagged end of the bolt. Trying to do this in situ is not easy. One way is to clamp a piece of steel to the component with a pilot hole already drilled in it to guide your drill bit centrally into the bolt.

If the bolt snapped while being removed because it was rusted into its thread then an Easy-Out is unlikely to work. Let's face it, if the full strength of the top half of the bolt wasn't enough to remove the threaded bit then a smaller tool fitted into a little pilot hole isn't going to help either. Chances are you'll break the Easy-Out inside the bolt and end up in an even worse mess. The way out here is to mill the bolt out carefully and retap the threads in the component or to helicoil the component back to its original size if the threads are too damaged to be used. Spark erosion is a useful method but expensive and obviously needs the component to be taken to a specialist. Welding a nut to any broken stub that can be reached is a good plan as described above.

If you don't have the tools or techniques to safely remove a broken bolt then don't try. You'll just damage things further. Take it to a machine shop and have it done properly.

Broken bolt Studs - The Two Nut Trick

To remove studs don't use stud extractors. They chew up the threads and destroy the studs as they remove them. Wind on two nuts and do them up tightly against each other. Use a spanner on the bottom nut and wind the stud out. If it won't come out with normal force then apply the hammer technique first :) this saves on the price of the stud or thread extractor .