  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 its
projection from a general point $P$ to a plane is a complete intersection of
curves of degrees $a$ and $b$. Examples which we call grids have been known
since 2011. The only nongrid nondegenerate examples previously known had
$ab=12, 16, 20, 24, 30, 36, 42, 48, 54$ or $60$. Here, for any $4 \leq a \leq
b$, we construct nongrid nondegenerate $(a,b)$-geproci sets in a systematic
way. We also show that the only such example with $a=3$ is a $(3,4)$-geproci
set coming from the $D_4$ root system, and we describe the $D_4$ configuration
in 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 interesting
properties. We also explore the relation between unexpected cones and geproci
sets and introduce the notion of $d$-Weddle schemes arising from special
projections of finite sets of points. This work initiates the exploration of
new perspectives on classical areas of geometry. We formulate and discuss a
range of open problems in the final chapter.
