We recall Vere-Jones's definition of the $alpha$--permanent and describe the connection between the (1/2)--permanent and the hafnian. We establish expansion formulae for the $alpha$--permanent in terms of partitions of the index set, and we use these to prove Lieb-type inequalities for the $pmalpha$--permanent of a positive semi-definite Hermitian $n imes n$ matrix and the $alpha/2$--permanent of a positive semi-definite real symmetric $n imes n$ matrix if $alpha$ is a nonnegative integer or $alphage n-1$. We are unable to settle Shirai's nonnegativity conjecture for $alpha$--permanents when $alphage 1$, but we verify it up to the $5 imes 5$ case, in addition to recovering and refining some of Shirai's partial results by purely combinatorial proofs.