Easy to state open problems in computability theory
One famous open question about the poset $(D, \leq_T)$ of Turing degrees is whether it has any non-trivial automorphisms. That is, does there exist a non-identity bijection $f\colon D \to D$ such that $a \leq_T b $ if and only if $f(a) \leq_T f(b)$?.