During the past year, many new contests have arised, some of which include the IPhOO, IChOO, JOMO, and Proofathon. There are many well-known contests, including (but not limited to) MATHCOUNTS, AMCs, etc.
On the other side, I'm sure there are other contests that I (and many others) have not heard of, which is the purpose of this thread.
Please post academic contests here, under the appropriate category in the comments section, as well as give them an introduction (which may be copied directly from the website!)
If you believe I have missed a topic, feel free to start a comment under that section.