The integrity of a graph , denoted , is defined by where denotes the maximum order of a component of ; further an -set of is any set for which the minimum is attained. Firstly some useful concepts are formalised and basic properties of integrity and -sets identified. Then various bounds and interrelationships involving integrity and other well-known graphical parameters are considered, and another formulation introduced from which further bounds are derived. The paper concludes with several results on the integrity of circulants.