The discovery of topological insulators has sparked considerable interest in the study of topological phases of matter. The hallmark of these novel phases is the emergence of topologically protected boundary phenomena, e.g. quantum pumping, surface states related to exotic models from particle physics and quasi-particles with non-Abelian statistics. Yet, realizations of these phases of matter are scarce, and moreover, one-dimensional systems without any additional symmetries are considered to lack such topological phenomena. Here we show that quasicrystals possess high-dimensional topological phases. We demonstrate that one-dimensional quasicrystals exhibit topological properties that were, thus far, thought to be limited to two-dimensional systems. Using photonic quasicrystals, we observe localized boundary states, which manifest these topological properties. These topological boundary states are used to realize an adiabatic optical pumping effect in which photons are unidirectionally pumped across the sample. This work opens a path to the realization of new types of topological phases, and provides new insights to the physics of quasicrystals.