Optimal searches for a fixed object are discussed and the rigorous analytical results of discrete search theory are presented. They show that the totally optimal, the uniformly optimal, the locally optimal, and the fastest searches are identical under not too restrictive assumptions. The mathematical formalism is illustrated by an Earth-approaching asteroid search and optimal searches for such objects are explicitly constructed. The approximation that Earth-approaching asteroids are fixed is equivalent to having a very high (>or=100 square degrees/hr) search rate. Generalizations to other types of astronomical search are briefly mentioned.