Abstract: I will describe several natural questions that arise in connection with the sequence of best approximation vectors. This sequence, which I will introduce in the talk, is a higher dimensional generalization of continued fraction convergents (which I will also discuss briefly). The questions involve the statistical behavior of certain observables, and the means to understand them involve a mix of number theory, ergodic theory, and elementary geometry. The talk will survey a lot of classical work on this topic, and the new results described are joint with Uri Shapira.