Abstract: There is a unique line through two points in the (complex projective) plane, and a unique conic through five points in general position (i.e., no three of them on a line). Classical algebraic
geometry shows that given four general lines in 3-space, there are exactly two lines intersecting all of them. These are easy examples of curve counting problems. We give an outline of the problems one wants to study and highlight the need to choose a (suitable) compactification and of finding the dimension of the corresponding moduli space; we then show how in some cases there are components of different dimensions and introduce the notion of expected dimension.