The azimuth moveout (AMO) operator, unlike the DMO operator, has a 3-D structure in homogeneous isotropic media, with an out-of-plane (crossline) component. In general, this component is concaved downward giving the operator an overall skewed-saddle shape. The AMO operator is typically smaller in size than conventional DMO operators. When velocity varies vertically, the operator shape changes depending on how the velocity varies. The general shape of the operator, however, remains overall saddle. In fact, for smooth velocity increases with depth, similar to those found in the Gulf of Mexico, the AMO operator does not vary much from its homogeneous counterpart. The residual AMO operator, constructed by cascading a forward homogeneous AMO operator with an inverse v(z) one is extremely small, which suggests that the impact of such v(z) variations on the AMO operator is generally small. Complex vertical velocity variations, on the other hand, result in more complicated AMO operators that include, among other things, triplications at moderate angles. Regardless of the complexity of the model, the v(z) operator has the same first order behavior as its homogeneous counterpart. As a result, for small dip angles the homogeneous AMO, as a tool for partial stacking, often enhances the image. Moderate to steep dips in complex v(z) media requires the application of an algorithm that honors such velocity variations.