It is common for nonlinear dynamical systems to have more than one stable time-asymptotic final state (i.e., more than one attractor). In these cases, the attractor to which an orbit tends will depend on its initial condition. The set of initial conditions that approach a given attractor is the basin of attraction for that attractor. A point is in the basin boundary if every t-neighborhood of it contains points in a t least two basins. Recently, there have been a number of papers investigating the properties of basin boundaries. In particular, basin boundaries have been shown to sometimes possess fractal structure;14 thus, such basin boundaries are called fractal basin boundaries. In such cases, a convenient characterization of the basin boundary is in terms of the basin boundary dimension dim (S n B), where S denotes some open set in phase-space that contains part of the basin boundary B (e.g., S might be a volume bounded by a smooth closed surface), and dim (S n B) is the dimension of the part of the basin boundary that lies in S. The dimension of the basin boundary dim (S n B ) can be defined in different ways, and an appropriate choice of definition can often be made to reflect issues of practical interest in a given situation. This is discussed further in the next sect ion. Fractal basin boundaries can have important practical consequences. In particular, for the purposes of determining which attractor eventually captures a given orbit, the arbitrarily fine-scaled structure of fractal basin boundaries implies that sensitivity to small errors in initial conditions can be vastly increased as compared to the situation where the boundary is not fractal. The sense in which this is true is discussed in references 1 and 2, where it is also shown that the boundary dimension dim (S n B) in S provides a quantitative measure of this sensitivity.