Let $G$ be reductive over the field $F$.
When $F$ is finite, there is Deligne-Lusztig theory.
When $F$ is archimedean, there is Langlands classification. (Knapp's book "Representation theory of semisimple groups tries to develop it).
When $F$ is supercudpidal and $G$ is $GL(n)$, $SL(n)$, then Bushnell and Kutzko have constructed the unitary dual explicitly.
For general $G$ and non-archimedean $F$, there is no construction of all supercuspidals known: see e.g. here for further references.
For general $F$ without trivial topology, you get discrete groups. To classify their unitary representation up to equivalence is hopeless. So you rather want to consider algebraic representations here. The smooth admissible/unitary representation theory of a local field is automatically algebraic in some sense.
Last but not least, a theory of zeta integrals would require Fourier-selfdual additive group of the field. This property characterizes local and finite fields among all fields.
I can't say much about the geometric Langland's program. Uses different fields and seems pretty algebraic to me, somebody else?