  Red-giant stars are low- to intermediate-mass ($M \lesssim 10$~M$_{\odot}$)
stars that have exhausted hydrogen in the core. These extended, cool and hence
red stars are key targets for stellar evolution studies as well as galactic
studies for several reasons: a) many stars go through a red-giant phase; b) red
giants are intrinsically bright; c) large stellar internal structure changes as
well as changes in surface chemical abundances take place over relatively short
time; d) red-giant stars exhibit global intrinsic oscillations. Due to their
large number and intrinsic brightness it is possible to observe many of these
stars up to large distances. Furthermore, the global intrinsic oscillations
provide a means to discern red-giant stars in the pre-helium core burning from
the ones in the helium core burning phase and provide an estimate of stellar
ages, a key ingredient for galactic studies. In this lecture I will first
discuss some physical phenomena that play a role in red-giant stars and several
phases of red-giant evolution. Then, I will provide some details about
asteroseismology -- the study of the internal structure of stars through their
intrinsic oscillations -- of red-giant stars. I will conclude by discussing
galactic archaeology -- the study of the formation and evolution of the Milky
Way by reconstructing its past from its current constituents -- and the role
red-giant stars can play in that.
