Abstract. I like the problem of dense packing of equal balls in the n-dimensional Euclidean space because of the abundance of mathematical disciplines connected to it. In the first half of the talk, I plan to give a survey of the modern state of the art, starting from small n and going upwards. In the second part I’ll show how algebraic geometry and number theory give us constructions of dense packings in large dimensions, focusing on the parallelism between number fields and curves over finite fields.