This is a very broad question and might be closed for that reason.

Anyways, one problem about decidability that continues to attract a lot of attention is extensions of [Hilbert's 10th problem][1] to other rings of number-theoretic interest, especially the rationals $\mathbb{Q}$. See for instance this nice [survey paper][2] of Poonen.


  [1]: https://en.wikipedia.org/wiki/Hilbert%27s_tenth_problem
  [2]: https://math.mit.edu/~poonen/papers/aws2003.pdf