New bideterminantal formulas for the irreducible symplectic and orthogonal characters are given that generalize the classical bideterminantal formulas. These formulas are analogous to Regev’s (Israel J. Math. 80 (1992), 155–160) bideterminantal formulas for Schur functions, the irreducible general linear characters. Also, new bideterminantal formulas for Proctor’s intermediate symplectic characters are derived.