  The standard axioms of set theory, the Zermelo-Fraenkel axioms (ZFC), do not
suffice to answer all questions in mathematics. While this follows abstractly
from Kurt G\"odel's famous incompleteness theorems, we nowadays know numerous
concrete examples for such questions. A large number of problems in set theory,
for example, regularity properties such as Lebesgue measurability and the Baire
property are not decided - for even rather simple (for example, projective)
sets of reals - by ZFC. Even many problems outside of set theory have been
showed to be unsolvable, meaning neither their truth nor their failure can be
proven from ZFC. A major part of set theory is devoted to attacking this
problem by studying various extensions of ZFC and their properties. We outline
some of these extensions and explain current obstacles in understanding their
impact on the set theoretical universe together with recent progress on these
questions and future scenarios. This work is related to the overall goal to
identify the "right" axioms for mathematics.
