Go was solved up to 5x6 only as of 2009. This table by Erik van der Werf and Mark Winands shows how many points komi to give white such that the solution is a draw for boards up to that size. For instance, on a 5x6 board White should be given 4 points; on a 5x5, White should be given 25 points.
Notice that 25 points is the size of the 5x5 board; ergo, White cannot make a living group on a 5x5 board given optimal Black play (which, I'm not positive, but I think can begin with play in the center). With the 25-point komi, using territory scoring, it becomes optimal play for Black to begin by passing, as it is the case that if Black places a stone, White can win by passing.
I would imagine that with the current state of algorithms and processing power, we're not too far off from solutions for 6x6 and 7x5 Go. But to give a sense of how the problem scales, their 2009 paper about solving 5x6 Go states:
The question now is when would 6×6 be solved? In 6 years time we went from a surface
of 25 (i.e., 5×5) to a surface of 30 (i.e., 5×6) of solved Go boards. This was not
only because of better hardware but also due to a better search engine.
We can try to predict when MIGOS II would be able to solve 6×6 in a reasonable amount
of time by extrapolating the current results....
An optimistic extrapolation suggests that on current hardware MIGOS II would require a
few years to solve 6×6. However, we could easily be underestimating by a factor of
100. Nevertheless, we believe that with some effort αβ-based solvers, such as MIGOS
II, should be able to solve 6×6 within the next 5 years, especially because signiﬁcant
improvements in the evaluation function are still possible.