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We call a set of points $Z\subset{\mathbb P}^{3}_{\mathbb C}$ an$(a,b)$-geproci set (for GEneral PROjection is a Complete Intersection) if itsprojection from a general point $P$ to a plane is a complete intersection ofcurves of degrees $a$ and $b$. Examples which we call grids have been knownsince 2011. The only nongrid nondegenerate examples previously known had$ab=12, 16, 20, 24, 30, 36, 42, 48, 54$ or $60$. Here, for any $4 \leq a \leqb$, we construct nongrid nondegenerate $(a,b)$-geproci sets in a systematicway. We also show that the only such example with $a=3$ is a $(3,4)$-geprociset coming from the $D_4$ root system, and we describe the $D_4$ configurationin detail. We also consider the question of the equivalence (in various senses)of geproci sets, as well as which sets occur over the reals, and which cannot.We identify several additional examples of geproci sets with interestingproperties. We also explore the relation between unexpected cones and geprocisets and introduce the notion of $d$-Weddle schemes arising from specialprojections of finite sets of points. This work initiates the exploration ofnew perspectives on classical areas of geometry. We formulate and discuss arange of open problems in the final chapter.

Configurations of points in projective space and their projections