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The display of surfaces and solids has usually been restricted to the domain of scientific visualization; however, little work has been done on the visualization of surfaces and solids of dimensionality higher than three or four. Indeed, most high-dimensional visualization focuses on the display of data points. However, the ability to effectively model and visualize higher dimensional objects such as clusters and patterns would be quite useful in studying their shapes, relationships, and changes over time. In this paper we describe a method for the description, extraction, and visualization of N-dimensional surfaces and solids. The approach is to extend generalized cylinders, an object representation used in geometric modeling and computer vision, to arbitrary dimensionality, resulting in what we term Generalized Hyper-cylinders (GHCs). A basic GHC consists of two N-dimensional hyper-spheres connected by a hyper-cylinder whose shape at any point along the cylinder is determined by interpolating between the endpoint shapes. More complex GHCs involve alternate cross-section shapes and curved spines connecting the ends. Several algorithms for constructing or extracting GHCs from multivariate data sets are proposed. Once extracted, the GHCs can be visualized using a variety of projection techniques and methods toconvey cross-section shapes.