I recently did a search for open problems in mathematics. Here are a few links I’ve found (manly via this math tackexchange post: https://math.stackexchange.com/questions/1354028/database-of-unsolved-problems-in-mathematics):

• List of unsolved problems in mathematics (on Wikipedia)
  This is a very long list of open problems, nicely categorized by area and with loads of references at the bottom.
• Unsolved Problems in Group Theory. The Kourovka Notebook
  This is living document on problems in Group Theory. It started in 1965(!) from a Russian conference in Kourovka that gave it its name, and has been updated ever since. I found the link to this via the above Wikipedia article.
• There is also this dedicated category page on Wikipedia with unsolved problems in number theory.
• http://www.openproblemgarden.org/
  A wiki of open problems with some annotations. It seems to be hosted by a university in Prague, Czech Republic.
• https://www.scilag.net/problems
  This seems to be an online service similar to the one above, but looking like it has been specifically developed to support communication on research problems in mathematics. They also have an about page which says:

SciLag is a free web-based platform for facilitating dynamic organisation of scientific problems at a research level. It represents an online service for scientists for sharing their knowledge about the forefront of research. The current version is set up for the Mathematical community only to test and refine the concept in action.
[…] The platform is designed to be home for scientific problems which are (were) open. For many of these open problems to be addressed, the science, mathematics in this case, needs new tools and ideas to be discovered. In other words, the current knowledge is lagging behind, hence SciLag for “Science Lagging behind”.

https://www.scilag.net/about-us