Particle accelerators are indispensable tools for probing matter at the smallest scales and the improvements of such tools depend on the progress and understanding of accelerator physics. The Compact Linear Collider (CLIC) is a proposed, linear electron–­positron collider on the TeV-scale, based at CERN. In such a large accelerator complex, diagnostics and alignment of the beam are crucial in order to maintain beam quality and luminosity. In this thesis we have utilized the nonlinear fields from the octupole component of the radio-frequency fields in the CLIC accelerating structures for beam-based diagnostics. We have investigated methods where the nonlinear position shifts of the beam are used to measure the strength of the octupole component and can also be used for alignment. Furthermore, from the changes in transverse beam profile, due to the nonlinear octupole field, we determine the full transverse beam matrix, which characterizes the transverse distribution of the beam. In circular accelerators, nonlinear fields result in nonlinear beam dynamics, which often becomes the limiting factor for long-term stability. In theoretical studies and simulations we investigate optimum configurations for octupole magnets that compensate amplitude-dependent tune-shifts but avoid driving fourth-order resonances and setups of sextupole magnets to control individual resonance driving terms in an optimal way.