CNN Money discusses some of the problems with nuclear waste disposal. Nuclear waste is presently disposed of on site at reactors in deep pools of water. Industry experts say this is a viable solution for a few decades, but a geological repository needs to be found - i.e. the waste needs to be buried deep in a geologically stable location.
Nuclear waste can be recycled, extracting the plutonium produced. However, this does come uncomfortably close to producing weapons-grade plutonium (reactor-grade plutonium or uranium cannot produce a viable nuclear weapon). Additionally, it will get rid of only half the waste (or seven eights at an optimistic estimate).