The main goal in disposing nuclear waste is to protect people and the environment. The majority of nuclear waste can be classified as low-level radioactive meaning it has low levels of radioactivity per volume or mass, i.e. protective suits that may be lightly contaminated.
As the radioactive nature of the waste decreases over time, if there was a place to store the waste until its components have decomposed, this would be an easy problem to solve. The issue is that the radioactive half-life of the materials can mean huge periods of time up to hundreds of thousands of years.
Most nuclear waste is still stored on the property of the nuclear plant that created it. Nuclear fuel is stored at the nuclear power plants where it is created in steel-lined concrete pools, water-filled basins, or huge airtight steel or concrete and steel containers with the ultimate goal of it being recycled by the government at some point in time.
Other options have been discussed: sub-seabed disposal, very deep-hole disposal, space disposal, ice-sheet disposal, island geologic disposal, deep-well injection disposal, and, of course, reprocessing or recycling the spent nuclear fuel. Each of these has its own advantages and disadvantages.
Currently, as stated on the website for the Nuclear Energy Institute, the United States "does not currently recycle used nuclear fuel" for "economic and national security reasons." However the federal government does intend on evaluating fuel cycles and the benefits and availability of advanced recycling technologies. The hope is that the used nuclear waste can become new fuel for nuclear energy.
Yucca Mountain, Nevada, has been studied for a nuclear waste repository since 1978. It has been recommended by the Department of Energy due to its location at the Nevada Test Site in Nye County, Nevada, design elements and natural barriers contained within the site itself. In June 2008, the U.S. Department of Energy filed a license application to the Nuclear Regulatory Commission. The final decision has not yet been made.