The international community is in agreement that a lepton collider in the TeV centre of mass energy range is required to leverage discoveries made at the Large Hadron Collider and expand the physics programme. Two future colliders are proposed. The International Linear Collider (ILC) will collide electron and positron bunches at a centre of mass energy of 500 GeV, upgradable to 1 TeV. The Compact Linear Collider (CLIC) is designed to reach 3 TeV.This thesis investigates the wakefields, which degrade the beam quality, and beam dynamics in the main linacs of the ILC, presenting the first direct comparison of beam dynamics for linacs made up of the alternative high gradient superconducting cavity designs - the Reentrant and Ichiro cavities. Higher order modes of the electromagnetic field in the cavities, which will be excited by the passage of the bunches, are calculated using finite difference and finite element techniques. A trapped dipole mode in the Ichiro cavity at 2.4498 GHz is identified.These modes are used as the basis for the beam dynamics studies. These simulations have demonstrated that ILC linacs made up of the new high gradient cavities, with targeted damping, would meet wakefield requirements for delivering high quality beams for particle physics studies. This result is important since any upgrade of the ILC from 500 GeV to 1 TeV centre of mass energy would make use of one of these high gradient cavity designs in the extension to the linacs.Beam dynamics in the CLIC beam delivery system (BDS), are also detailed. Simulations included deflecting mode Crab Cavities required to maximise collision luminosity when there is a crossing angle, and verify analytic results for the required deflecting voltage and tolerances to phase differences. The tolerance to crab cavity roll angle is found to be extremely tight, at 5.9 millidegrees. Additionally, results in this thesis uncover a problem with the BDS magnet layout which must be addressed.