It is well known that the imposition of a constraint can transform the properties of critical systems. Early work on this phemomenon by Essam and Garelick, Fisher, and others, focused on the effects of constraints on the leading critical exponents describing phase transitions. Recent work extended these considerations to critical amplitudes and to exponents governing logarithmic corrections in certain marginal scenarios. Here these old and new results are gathered and summarised. The involutory nature of transformations between the critical parameters describing ideal and constrained systems are also discussed, paying particular attention to matters relating to universality.