The term \emph{shrinking target problems} in dynamical systems describes a class of questions which seek to understand the recurrence behavior of typical orbits of a dynamical system. The standard ingredients of such questions are a probability measure preserving dynamical system $(X,\mu, T)$ and a sequence of targets $\{B_m\}_{m\in \mathbb{N}}$ with $B_m\subset X$ and $\mu(B_m)\to 0$. Classical questions in this area focus on the set of points whose $n$-th iterate under $T$ lies in the $n$-th target for infinitely many $n$. We study a uniform analogue of these questions, the so-called \emph{eventually always hitting points}. These are the points whose first $n$ iterates will never have empty intersection with the $n$-th target for sufficiently large $n$. We search for necessary and sufficient conditions on the shrinking rate of the targets for the set of eventually always hitting points to be of full or zero measure. In particular, I will present some recent dichotomy results for some interval maps obtained in collaboration with Mark Holland, Maxim Kirsebom, and Tomas Persson. We highlight connections to extreme value theory.