The authors apply a theorem of J. Lurie to produce cohomology theories associated to certain Shimura varieties of type $U(1,n-1)$. These cohomology theories of topological automorphic forms ($mathit{TAF}$) are related to Shimura varieties in the same way that $mathit{TMF}$ is related to the moduli space of elliptic curves. Table of Contents: $p$-divisible groups; The Honda-Tate classification; Tate modules and level structures; Polarizations; Forms and involutions; Shimura varieties of type $U(1,n-1)$; Deformation theory; Topological automorphic forms; Relationship to automorphic forms; Smooth $G$-spectra; Operation on $mathit{TAF}$; Buildings; Hypercohomology of adele groups; $K(n)$-local theory; Example: chromatic level $1$; Bibliography; Index. (MEMO/204/958)