Nuclear energy has been a hot-button issue for a very long time, both domestically and internationally. It provides a significant portion of the global electricity supply, as you will see in the image below.
Nuclear energy is non-renewable. Uranium is by far the most-used nuclear fuel. As with other non-renewable fuels, all of the uranium that is on earth now is all that we will ever have, and estimates can be made of the remaining recoverable resources. At current rates of consumption, we will not run out of uranium any time soon. But this depends very highly on a number of variables, including keeping consumption at current levels, technology not advancing, estimates of reserves changing, and so forth.
The World Nuclear Association (WNA), an industry association, provides a very thorough explanation of possible complicating factors [3], but they state that at current rates of consumption, the world has enough reserves to last about 90 years. The Nuclear Energy Agency (NEA [4]), like the WNA, [4]is effectively an industry group and has a wealth of expertise at its disposal. They indicate that as of 2009, the world had about a 100 year supply of uranium. So it appears that as long as the rate of use does not increase there is a little less than 100 years of nuclear fuel supplies left.
According to the World Nuclear Association, there are 454 operable reactors worldwide with a further 54 under construction [5]. The technology is well known by now, and despite the extreme danger posed by nuclear meltdowns, there have been very few major incidents. You are probably familiar with the Fukushima Daichi meltdown that happened in 2011, and perhaps heard of Chernobyl in Ukraine in 1986 (still the worst nuclear disaster to date), and maybe even Three Mile Island in the U.S. in 1978. Here is a partial list of nuclear accidents [6] in history from the Union of Concerned Scientists (UCS). But putting aside this risk at the moment, nuclear energy has shown itself to be a viable source of electricity, and likely will continue to be used for the foreseeable future. Among other things, nuclear power plants generally have a useful lifetime of around 40-60 years, so we are "locked in" until mid-century at least.
Nuclear energy is a mixed bag in terms of the question of sustainability. You may recall that nuclear is considered a carbon-free source, and since it is a proven and reliable source, it is seen by many as a good option. Note that despite being considered "carbon-free," nuclear energy results in some lifecycle emissions because of the materials used in mining, building the power plant, and so forth. (Lifecycle emissions are all the emissions generated by all processes required to make an energy source, including things like mining of materials, manufacturing of equipment, and operating equipment.) But according to the National Renewable Energy Laboratory (NREL) [7] it has approximately the same lifecycle emissions as some renewable energy sources.
Some other sustainability considerations include:
The first article below is a good example of why it pays to pay attention to citations and be well informed on a topic, in regards to finding good information sources. The article is on a website that I've never heard of before, so at first, I was suspicious of the content. However, they provide legitimate sources for the information presented, and I have enough prior knowledge to know that the arguments they put forth are legitimate. Overall, it's a good summary of some of the pros and cons of nuclear energy.
Overall, nuclear is reliable and almost carbon-free but is expensive and non-renewable. Also, because power plants are so expensive to build, once they are built they are generally used for as long as possible, as long as they are still economic. When accidents happen, they can be catastrophic, but they are extremely rare. However, the waste product from nuclear power plants is dangerous for thousands of years, and right now we have no way of safely disposing of it - it is kept in storage, usually at the power plants themselves.
Links
[1] https://www.e-education.psu.edu/emsc240/node/571
[2] http://www.tsp-data-portal.org/Breakdown-of-Electricity-Generation-by-Energy-Source#tspQvChart
[3] http://www.world-nuclear.org/info/Nuclear-Fuel-Cycle/Uranium-Resources/Supply-of-Uranium/
[4] http://www.oecd-nea.org/
[5] http://www.world-nuclear.org/Nuclear-Basics/Global-number-of-nuclear-reactors/
[6] http://www.ucsusa.org/nuclear-power/nuclear-power-accidents/history-nuclear-accidents#.Veeuk_lViko
[7] https://www.nrel.gov/docs/fy13osti/57187.pdf
[8] http://www.nrc.gov/reading-rm/doc-collections/fact-sheets/radwaste.html
[9] https://www.epa.gov/navajo-nation-uranium-cleanup
[10] http://www.eia.gov/outlooks/aeo/pdf/electricity_generation.pdf
[11] https://understandsolar.com/pros-and-cons-of-nuclear-energy/
[12] http://www.npr.org/2016/04/07/473379564/unable-to-compete-on-price-nuclear-power-on-the-decline-in-the-u-s
[13] http://www.nytimes.com/2014/12/23/science/nuclear-carbon-free-but-not-free-of-unease-.html?_r=0
[14] http://cen.acs.org/articles/91/web/2013/04/Nuclear-Power-Prevents-Deaths-Causes.html