We examine the question of when a polynomial f over a commutative ring has a nontrivial functional decomposition j = g ° h. Previous algorithms are exponential-time in the worst case, require polynomial factorization, and only work over fields of characteristic 0. We present an 0(n^2)-time algorithm, where r is the degree of g. We also show that the problem is in NC. The algorithm does not use polynomial factorization, and works over any commutative ring containing a multiplicative inverse of r. Finally, we give a new structure theorem that leads to necessary and sufficient algebraic conditions for decomposibility over any field. We apply this theorem to obtain an NC algorithm for decomposing irreducible polynomials over finite fields, and a subexponential algorithm for decomposing irreducible polynomials over any field admitting efficient polynomial factorization.