Before attempting to remove a broken bolt using any of the methods below you must ensure that you wear the correct PPE, so safety glasses and gloves should be worn at all times. When welding a welding helmet and gauntlets should be worn. Also please see the article on welding safety
Removing a broken bolt or stud using a welder can work well as the heat generated during the welding process can help to release the bond between the threads.
Here we have a broken bolt that is almost flush with the surface of the metal plate
One of the best ways to remove a broken bolt like this is by welding a nut onto the top of the bolt, here we are using an arc welder to weld the nut to the top of the bolt.
A mig welder is equally as good if not better! It's important to use the correct size nut and ensure that it is dead over the threaded part of the bolt.
Make sure the welder is on high amps as the heat generated during the welding process can help to loosen a stubborn bolt.
Here you can see that the intense heat generated has made the nut glow, this heat can help to loosen a stuck bolt.
Allow the bolt to cool, for a few seconds (remember it will be extremely hot for a long time) then try and remove the bolt using a suitable spanner.
Then place the hot bolt somewhere safe (out of the reach of people and pets etc) until it cools.
For bolts or studs that have sheared deeper inside of the hole you can get some special rods called X-Tractalloy
You can also try these two methods- Remove a broken bolt using a left handed drill bit | Easy out | screw extractor