  The study of inner models was initiated by G\"odel's analysis of the
constructible universe. Later, the study of canonical inner models with large
cardinals, e.g., measurable cardinals, strong cardinals or Woodin cardinals,
was pioneered by Jensen, Mitchell, Steel, and others. Around the same time, the
study of infinite two-player games was driven forward by Martin's proof of
analytic determinacy from a measurable cardinal, Borel determinacy from ZFC,
and Martin and Steel's proof of levels of projective determinacy from Woodin
cardinals with a measurable cardinal on top. First Woodin and later Neeman
improved the result in the projective hierarchy by showing that in fact the
existence of a countable iterable model, a mouse, with Woodin cardinals and a
top measure suffices to prove determinacy in the projective hierarchy. This
opened up the possibility for an optimal result stating the equivalence between
local determinacy hypotheses and the existence of mice in the projective
hierarchy. This article outlines the main concepts and results connecting
determinacy hypotheses with the existence of mice with large cardinals as well
as recent progress in the area.
