Recent publications on NP ?= coNP question
NP is equal to coNP if and only if there are efficiently verifiable proofs of unsatisfiability. I.e., if and only if there exists a polynomial time turing machine $M$, which given any SAT formula $\phi$ and a string $\pi$ outputs $M(\phi, \pi) = 1$ if and only if $\phi$ is unsatisfiable. Most theorists believe there are no such efficient proofs, but proving that they don't exist would resolve the P vs NP question. However, there has been progress in showing that proofs of a restricted type must necessarily be superpolynomial in size. This is the subject of proof complexity: see the foundational paper by Cook and Reckhow, the  survey by Krajicek, or these lecture notes by Razborov.