In several areas of mathematics, including probability theory, statistics and asymptotic convex geometry, one is interested in high-dimensional objects, such as measures, data or convex bodies.  One common theme is to try to understand what lower-dimensional projections can say about the corresponding high-dimensional objects.   I will describe how this line of inquiry leads to geometric generalizations of some classical results in probability related to tails of random projections, both in commutative and non-commutative settings, and also discuss their relation to some long standing open problems in convex geometry.