Full-order observers have n integrators (or n delayors in the discrete-time case) that must be assigned ‘initial-condition values’ prior to the beginning of operation. However, the issue of how best to choose (design) those initial-condition values has apparently been completely ignored in control engineering textbooks. As a consequence, observer initial conditions in industrial applications are usually set to zero, by default. In this paper it is shown that there exists a welldefined, easily computed, physically realizable optimum choice for observer initial-condition values. When these optimum initial conditions are used, rather than ‘zero’ initial conditions, a significant improvement in the observer's ‘start-up error transients’ may result. A numerical example is worked out to illustrate this performance improvement.