A transverse multipole expansion is derived, including the longitudinal components necessarily present in regions of varying magnetic field profile. It can be used for exact numerical orbit following through the fringe-field regions of magnets whose end designs introduce no extraneous components, i.e., fields not required to be present by Maxwell's equations. Analytic evaluations of the deflections are obtained in various approximations. Mainly emphasized is a "straight-line approximation," in which particle orbits are treated as straight lines through the fringe-field regions. This approximation leads to a readily-evaluated figure of merit, the ratio of rms end deflection to nominal body deflection, that can be used to determine whether or not a fringe field can be neglected. Deflections in "critical" cases (e.g., near intersection regions) are analyzed in the same approximation.