We apply the theory of random walks to quantitatively describe the general problem of how to search efficiently for randomly located objects that can only be detected in the limited vicinity of a searcher who typically has a finite degree of "free will" to move and search at will. We illustrate Lévy flight search processes by comparison to Brownian random walks and discuss experimental observations of Lévy ights in the special case of biological organisms that search for food sites. We review recent findings indicating that an inverse square probability density distribution P(<img SRC="http:/img/fbpe/bjp/v31n1/sele.gif">) ~ <img SRC="http:/img/fbpe/bjp/v31n1/sele.gif">-2 of step lengths <img SRC="http:/img/fbpe/bjp/v31n1/sele.gif">can lead to optimal searches. Finally we survey the explanations put forth to account for these surprising findings.