I will discuss recent progress on two specific instances: Firstly, the construction of extensions with prescribed Galois group and "small" ramification indices. This is directly related to the construction of low-degree number fields with unramified $G$-extensions and leads to generalized Cohen-Lenstra heuristics.
Secondly, the construction of $G$-extensions with "powerfree" discriminant. This generalizes previous extensive investigations about fields with squarefree discriminants, corresponding to the special case $G=S_n$.
If time allows, I will also discuss some work in progress about Galois realizations with prescribed decomposition groups.