Nuclear power stations make energy by splitting uranium atoms, this is fission. But you can make evergy by joining light elements like Hydrogen to make a heavier one like Helium. This is fusion and is the way the sun makes energy.
But although the raw materials are abundant and fusion is inherently safer it's very hard to do.
As you'll see 2 positively charged nucleii are brought closer and closer together the electric force pushes them away with incredible strength. They have to get very very close before the strong nuclear force can grab them and the nucleii become fused.
This means the nucleii need to be moving fast - very fast and that means very hot, temperatures similar to the inside of the sun.
This has been achieved, the JET tokamak
(
http://en.wikipedia.org/wiki/Joint_European_To rus )
managed 16 MW although more energy had to be put in than that to get it to work. The next one ITER will "make a profit"
Cold fusion was first claimed in 1989 - it was claimed that small amounts of fusion had been seen simply by electrolysing heavy water with palladium electrodes.
This would be surprising as there's no known way of getting past the electric repulsion of the nucleii other than force so it basically contradicts known physics.
Two other teams in the US said they'd seen it too and then said "whoops sorry" and basically it all blew over.
There is however a continuing undercurrent of research, some firms like Toyota are playing the CF lottery and it regularly pops up in popular science programs.
Fusion will be running our children's world but I don't think it'll start in a test tube, it'll start here:
http://www.iter.org