I will talk about the problem of expressing permanent of matrices as
determinant of (possibly larger) matrices. This problem has close
connections with complexity of arithmetic computations: complexities of
computing permanent and determinant roughly correspond to arithmetic
versions of the classes NP and DP respectively. I will survey known
results about their relative complexity and describe two recently
developed approaches that might lead to a proof of the conjecture that
permanent can only be expressed as determinant of superpolynomial-sized
matrices.