We extend the Malliavin calculus from the classical finite variance setting to generalized processes with infinite variance and their test processes. The domain and the range of the basic Malliavin operators is characterized in terms of test processes and generalized processes. Various properties are proved such as the duality of the integral and the derivative in strong and in weak sense, the product rule with respect to ordinary and Wick multiplication and the chain rule in classical and in Wick sense.