The framework of the Weil conjectures establishes a correspondence between the arithmetic of varieties over finite fields and the topology of the corresponding complex varieties. Many varieties of interest arise in sequences, and a natural extension of the Weil conjectures asks for a relationship between the asymptotic point count of the sequence over finite fields and the limiting topology of the sequence over C.  In this talk, I'll recall the Weil conjectures and explain the basic idea of these possible extensions.  I'll then give a survey of ongoing efforts to understand and exploit this relationship, including Ellenberg-Venkatesh-Westerland's proof of the Cohen-Leinstra heuristics for function fields, a ``best possible'' form of this relationship in the example of configuration spaces of varieties (joint with Benson Farb), and a counterexample to this principle coming from classical work of Borel and recent work of Lipnowski-Tsimerman.