Nuclear fission breeder reactors are a possible alternative to conventional nuclear power. They could overcome some of its main problems.
The only breeder reactors that exist are experimental/research-based, not providing electricity on a commercial scale.
This means that fuel scarcity would no-longer be a reason why nuclear power can't scale up and replace fossil fuels. Breeder reactor fuels, while not renewable, have mineral reserves are far greater than coal, oil and gas combined (in terms of energy).
Uranium-238 & thorium-232 are not directly fissionable (unlike uranium-235), but they are fertile which means they can be "bred" to become fissionable materials. This where the term "breeder reactor" comes from.
For this process to work, breeder reactors still need to start with some fissile material, but they produce more of it than what they started with. A breeder reactor's doubling-time is a useful way to measure this: how long it takes for the reactor to produce twice as much fissile material as it started with. Breeder reactors so far have a doubling-time of over 100 years, unfortunately.  Scientists are trying to get this down to 10 years (no guarantees). Even in such a case, it would probably still take decades to breed enough fuel to meet global energy demands (unless uranium/thorium mining could somehow be done at an extremely fast rate?[RESEARCH needed]).
Risk of weapons proliferation
Uranium-238 breeder reactors are high risk. The uranium-238 is bred to become plutonium-239, which is considered "easy" to make nuclear bombs with.
Thorium-232 breeder reactors are lower risk. The thorium-232 is bred to become uranium-233, which is considered far more difficult for making nuclear bombs.
If breeder reactors are to become a mainstream source of energy, they have to be designed extremely securely[ELABORATION needed] to prevent the possibility of anyone maliciously siphoning off some of the fissionable materials. [RESEARCH needed]
Breeder reactors produce less than 1/1000th as much nuclear waste as conventional nuclear power
- Breeder reactor - Energy Education - another wiki which elaborates more on the technical details
- Fast Neutron Reactors
- Fast breeder reactors
Some reactor types that might show potential
- Traveling wave reactor
- Molten salt reactor (Thorium)
- Liquid metal fast breeder reactors (LMFBR) (Uranium-238)
- Supercritical-Water-Cooled Reactor (SCWR)
- Example quoted from energyeducation.ca:
Thorium hasn't been used in large scale reactors, however some reactors have used it successfully in the past. A light water breeder reactor in Shippingport, Pa. USA operated for 5 years, and by the end of its operation it had 1.4% more fissile fuel than it began with.
This cites another source:
World Nuclear Association. (June 19 2015). Thorium [Online], Available: http://www.world-nuclear.org/info/Current-and-Future-Generation/Thorium/#b