Problems with interesting, non-trivial analogues in finite fields
The classification of simple Lie groups over $\mathbb R$ and $\mathbb C$ (Killing / Cartan) predated and, at least partly, inspired the classification of finite simple groups, which, at least to a great part, is made up of simple groups of Lie type over finite fields (Chevalley, Steinberg, Tits, Suzuki / Ree ...). Of course now it's exactly the ones that are not of Lie type which often get the limelight, but still ...
The Weil's conjectures can be seen as an analogue of the Riemann hypothesis for finite fields.