Scaling properties of domain growth in a variety of two-dimensional model systems were investigated by computer simulation techniques. Physical realizations of the models include molecular monolayers adsorbed on solid surfaces. Such systems are useful for studying general aspects of domain growth kinetics since they allow for a systematic variation of the symmetry and degeneracy of the ordering domains. Evidence is presented in favour of a dual universal classification of domain growth kinetics into classes of systems with soft and hard domain walls. The growth exponent for soft wall kinetics is smaller than the classical value, n equals one-half , of Lifshitz-Allen-Cahn curvature driven growth which is characteristic of hard wall kinetics.