Quantitative and Qualitative Aspects of Northcott's Theorem

About 60 years ago, Northcott has shown that sets of algebraic points in projective space of uniformly bounded degree and height are finite. However, asymptotic estimates, as the height gets large, are known only if the dimension is either one or large. We present some new results in this direction, and we will explain the basic proof line. Bombieri, Dvornicich, and Zannier also proposed the problem of extending Northcott's Theorem to other fields than numberfields. We will present a simple but quite general criterion that provides many new examples of such fields.