In his foundational 1932 work, Theorie des operations lineaires, Stefan Banach noted that the familiar Hilbert space $\ell^2$ was the only known example of a ``space of type (B)’’ (later, ``Banach space'') isomorphic to all of its closed infinite-dimensional subspaces. Whether any other such space exists became known as the ``homogeneous space problem’'. We will survey this and related questions from Banach space theory, as well as its 1996 resolution due to W. T. Gowers using deep ideas from infinitary combinatorics. Only a passing familiarity with Banach spaces, particularly $\ell^2$, will be assumed.