Wiring up the thermostat is usually pretty straightforward if an old one is being replaced. If a thermostat is being put in where one had never been before, that could involve running new wiring. The placement of the thermostat is also important so that its sensors can get an accurate reading of the temperature. A thermostat that is blocked by a bookcase or other large furniture will not get a good reading and will not perform efficiently.
Fuses -- Anyone who has worked with electrical systems knows all about fuses and how they fail. They can burn out over time, may just be loose, or can blow out during an electrical storm or due to overload from another failed component. Of course, that's what they're supposed to do; they stop surges from going through and damaging the rest of the system. When a fuse fails, whatever system it was protecting will stop working.
DIY: Most DIY installations go smoothly. The major brands have compatibility checkers to see which of their models will work with the wiring your old thermostat is using. Potential problems involve the need for a C-wire, or common wire, when the existing bundle doesn’t include one. There are solutions discussed in our Thermostat Buying Guide, linked to below.