This paper surveys the authors' and their collaborators' works on the attempt to unify the controllability theory of parabolic and hyperbolic equations. First of all, we show that the observability inequalities of the dual systems for those two types of equations of a different nature may be derived by means of a global Carleman-type estimate, which is based on a point-wise estimate for the corresponding principal operator. Next, we show that the null controllability of the heat equation may be obtained as the limit of the exact controllability of a family of singularly perturbed damped wave equations. Finally, we illustrate the complexity of the unification problem by analyzing a remarkable difference between the controllability of a class of hyperbolic-parabolic systems with control action entering the system through the wave component and the same problem but with control through the heat component.
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