Estimating causal effects in clinical trials is often complicated by treatment noncompliance and missing outcomes. In time-to-event studies, estimation is further complicated by censoring. Censoring is a type of missing outcome, the mechanism of which may be non-ignorable. While new estimates have recently been proposed to account for noncompliance and missing outcomes, few studies have specifically considered time-to-event outcomes, where even the intention-to-treat (ITT) estimator is potentially biased for estimating causal effects of assigned treatment. In this thesis, we develop a series of parametric potential-outcome (PPO) survival models, for the analysis of randomised controlled trials (RCT) with time-to-event outcomes and noncompliance. Both ignorable and non-ignorable censoring mechanisms are considered. We approach model-fitting from a likelihood-based perspective, using the EM algorithm to locate maximum likelihood estimators. We are not aware of any previous work that addresses these complications jointly. In addition, we give new formulations for the average causal effect (ACE) and the complier average causal effect (CACE) to suit survival analysis. To illustrate the likelihood-based method proposed in this thesis, the HIP breast cancer trial data \citep{Baker98, Shapiro88} were re-analysed using specific PPO-survival models, the Weibull and log-normal based PPO-survival models, which assume that the failure time and censored time distributions both follow Weibull or log-normal distributions. Furthermore, an extended PPO-survival model is also derived in this thesis, which permits investigation into the impact of causal effect after accommodating certain pre-treatment covariates. This is an important contribution to the potential outcomes, survival and RCT literature. For comparison, the Frangakis-Rubin (F-R) model \citep{Frangakis99} is also applied to the HIP breast cancer trial data. To date, the F-R model has not yet been applied to any time-to-event data in the literature.