The phase-field crystal model allows the study of materials on atomic length and diffusive time scales. It accounts for elastic and plastic deformation in crystal lattices, including several processes such as growth, dislocation dynamics, and microstructure evolution. The amplitude expansion of the phase-field crystal model (APFC) describes the atomic density by a small set of Fourier modes with slowly-varying amplitudes characterizing lattice deformations. This formulation is akin to a classical multiphase-field model for complex order parameters, describing interfaces and defects simultaneously. So far, the APFC framework has been used mostly to model basic lattice symmetries, such as triangular and square in two dimensions, and body-centered cubic and face-centered cubic in three dimensions. We present a coarse-grained description of quasicrystals, a unique state of matter lying between crystalline and amorphous structures. Their study requires understanding the interaction between the macroscopic properties of the material and the exotic microscopic arrangement, characterized by a lack of translational symmetry. Leveraging the multiphase field framework of the APFC model, we propose a novel mesoscale theory for quasicrystals. We characterize the topological defects forming in the structure, and we discuss their kinematics. Finally, we derive self-consistent laws for linear elasticity in the quasicrystal.