For the general MAX SAT problem, the currently best algorithm, is due to Asano and Williamson, available here . They show 0.7846-approximation. Note that there is still a gap between the lower and upper bound. MAX kSAT is the variant of MAX SAT where each cluase is made out of at most k variables. The problem of max-sat is related to constraint satisfaction and learning, but have a lot of other applications.
People currently can solve such instances when their size is relatively small.