In this talk, we will discuss some recents developments around two classical problems in discrete geometry, where projection theory plays an important role: the Erdős-Szekeres problem and the Heilbronn triangle problem. The former concerns the question of determining the size of largest convex subset in a given finite set of points, whereas the latter asks for the size of the smallest area triangle (also as a function of the number of points), if the point set is constrained to lie in a bounded region. Along the way, we will also be discussing several other old and new connections between these problems and various different parts of mathematics.