The disposal of nuclear waste is both an engineering and a political challenge. As far as the engineering side of that challenge is concerned, it basically boils down to developing ways to safely contain dangerous radioactive substances for several centuries, while the material decays to a harmless level of radiation. That seems like a long time, but most solutions accepted as effective by professionals in the field involve deep geological burial, and in geologic terms, a few centuries are like the blink of an eye. Finding stable geological formations for burial of radiological waste lends itself to straightforward analysis and should be a readily solved problem. However, the solution to this problem has been fraught in the USA with both engineering difficulties and resistance from local residents that have delayed the development of a permanent, deep geological repository.
The more difficult problem in the disposal of radioactive waste is the political challenge. Much resistance arises from local populations when an area is selected or undergoes examination for a repository site. Legitimate challenges to the engineering solution get mixed in with apprehension over having harmful radiological substances stored nearby and cause long delays in developing a safe repository for the material.