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.