Neutron sources provide a key tool in the study of materials and one of the key limiting factors in these experiments is often the total neutron flux. The European Spallation Source (ESS) in Lund, Sweden has set the goal to become the most intense source of cold neutrons in the world. At ESS a 2 GeV, 62.5 mA proton beam will be collided with a solid tungsten target to produce neutrons via spallation with an average beam power of 5 MW. The desired energy is obtained through the use of three families of superconducting accelerating structures from which 96% of the beam energy is gained. The fundamental goal in the accelerator design is to meet the desired power whilst minimising losses which can reduce the performance of the machine and may cause damage to the many sensitive components. One possible source of beam loss in the accelerator is beam-excited higher-order modes (HOMs). These are usually damped using HOM couplers to reduce the impact on the beam, at ESS however, designers have opted to forgo their use and rely instead on careful cavity design and production. Manufacturing errors are inherent in production processes and it is these which can result in the frequencies of HOMs varying from cavity-to-cavity---which in the worst case could have catastrophic consequences for the machine. The focus in this research is to analyse the impact of HOMs when manufacturing errors are present. To this end, detailed modal simulations have been performed to study the cavity designs and the impact of geometric errors on their modal spectra. These simulations have been used in conjunction with an equivalent circuit model to analyse the impact of geometric errors in individual cells of the full modal structure of the cavity. These simulations suggest that errors of less than 400 ÃÂµm are sufficient to prevent the HOMs in the elliptical cavities becoming dangerous. This has been combined with detailed beam dynamics studies performed using a drift-kick-drift scheme to analyse the limits set by ESS to mitigate the impact of HOMs on the beam. The result of this study was series of limits on the frequencies and R/Q of HOMs with the most important being a possible reduction in the allowable separation of HOMs from harmonics of the bunch frequency by up to 50%. In addition, a redesign of the high-beta cavity was undertook, which reduced the frequency separation of the dangerous HOMs from the ESS HOM frequency separation limit of 5 MHz from 5.38 MHz to 12.95 MHz.