Abstract: In many geometric algorithms we are given a family $F$ of geometric objects satisfying certain conditions and we wish to conclude that $F$ can be pierced by a ``small" set of points (and also construct such a piercing set). Such problems are sometimes called Gallai-type problems. In this talk we will survey the fundamental mathematical notions and tools used in tackling such problems. Specifically, we will explain the notion of Vapnik-Chervonenkis dimension (VC-dimension) of a hypergraph (an important notion in other areas such as computational learning theory, statistics, discrepancy theory) and the notion of epsilon-nets. Those notions play an important role in understanding the intersection patterns of many families of geometric objects. If time permit, we will also introduce the notion of weak-epsilon nets and its relation to the celebrated result of Alon and Kleitman from 1995 solving an important Helly-type conjecture of Hadwiger and Debrunner from 1957. The talk does not require any particular prior knowledge and is intended for a wide audience.