The continued fraction algorithm generates an integer sequence which terminates for a rational number, is periodic for the roots of irreducible integer quadratics, and is non-periodic for other algebraic numbers. This sequence uniquely determines the number in a useful way, e.g. one can compute convergents and solve Diophantine equations.

Does there exist a corresponding algorithm for, say, roots of irreducible cubics which has similar properties? What about other algebraic numbers? What is known about this? Or to save people time, what phrase should I google to find out the answer?