Given a variety over the rational numbers, a central objective in number theory is a good understanding of its set of rational points. For applications it is often useful to have upper bounds on counting rational points which are largely independent of the variety itself. For projective varieties, a general uniform upper bound was conjectured by Heath-Brown and Serre, and is now known in almost all cases by work of Browning, Heath-Brown and Salberger. I will give a general introduction to this topic of counting rational points, and discuss some recent work on dimension growth for affine varieties. This is partially joint work with Raf Cluckers, Pierre Dèbes, Yotam Hendel and Kien Nguyen.