We define large subalgebras of simple C*-algebras. These have several applications to the structure of crossed products, of which we concentrate on one: giving an upper bound on the radius of comparison (an entirely C*-algebraic invariant) of the crossed product by a minimal homeomorphism in terms of the mean dimension of the homeomorphism (a purely dynamical notion). Our result applies whenever the space involved has infinitely many connected components. We will define large subalgebras, and in particular give the main ideas of the proof that a large subalgebra has the same radius of comparison as the containing algebra. Some lecture notes for more extensive lectures on large subalgebras and the radius of comparison can be found at: http://pages.uoregon.edu/ncp/Courses/2015WyomingLargeSubalgs/2015WyomingLargeSubalgs.html