In this talk we introduce a gepro-$P$ property of sets of points $Z$ in projective space, meaning that the GEneral PROjection of $Z$ to a hyperplane has property $P$.

We focus on the geproci sets, i.e., when the property $P$ is "being a complete intersection". We present examples of geproci sets and a construction of nongrid geproci sets. We discuss recent results in this area, which started to be explored at the workshop on Lefschetz Properties and Jordan Type in Algebra, Geometry and Combinatorics in Levico Terme in 2018. We give a link between geproci sets, unexpected cones and the Weak Lefschetz Property.

The talk is based on a joint project with Luca Chiantini, Giuseppe Favacchio, Brian Harbourne, Juan Migliore, Tomasz Szemberg and Justyna Szpond.