The desire to understand the underlying geometry of multidimensional problems motivated several visualization methodologies to augment our limited 3-dimensional perception. After a short overview, Parallel Coordinates are rigorously developed obtaining a 1-1 mapping between subsets of Euclidean N-space and subsets of 2-space. It leads to representations of lines, flats, curves, intersections, hypersurfaces, proximities and geometrical construction algorithms. Convexity can be visualized in ANY dimension as well as non-orientability (Moebius strip) and other properties of hypersurfaces. The development is interlaced with several applications including Visual and Automatic Data Mining on real high dimensional datasets.