It is a common question in dynamics to ask how orbits of the system hit a target, i.e. a certain subset of the system. In the last couple of decades a new variation of this question emerged known as shrinking target problems. In this case we ask how orbits hit a sequence of sets of decreasing measure. Typical results in this direction are known as quantitative Poincaré recurrence, logarithm laws, dynamical Borel-Cantelli lemmas, hitting/return time statistics and extreme value distributions.
In this talk I will give an overview of these type of results, what information each result provides and how they are connected. I will also survey recent results in this direction. Finally, I will present own results on extreme value distributions for one-parameter subgroups acting on homogeneous spaces. For the purpose of the talk I will consider the concrete case of extremes for shortest vectors in lattices in SL(d,R)/SL(d,Z).