The irrational rotation algebras are the most studied and most accessible examples of 'non-commutative manifolds'. The authors investigate and classify smooth transformation groups on these algebras, using Lie algebraic techniques. The decisive question is the existence of representations of the Heisenberg Lie algebra by smooth 'vector fields'. They answer this, making use of the solution by Connes and Rieffel (1987) of the Yang-Mills problem for smooth 'vector bundles' over these algebras.