Indiana University-Purdue University Indianapolis (IUPUI) / Survival analysis has broad applications in diverse research areas. In this dissertation, we consider an innovative application of survival analysis approach to phase I dose-finding design and the modeling of multivariate survival data. In the first part of the dissertation, we apply time to event analysis in an innovative dose-finding design. To account for the unique feature of a new class of oncology drugs, T-cell engagers, we propose a phase I dose-finding method incorporating systematic intra-subject dose escalation. We utilize survival analysis approach to analyze intra-subject dose-escalation data and to identify the maximum tolerated dose. We evaluate the operating characteristics of the proposed design through simulation studies and compare it to existing methodologies. The second part of the dissertation focuses on multivariate survival data with semi-competing risks. Time-to-event data from the same subject are often correlated. In addition, semi-competing risks are sometimes present with correlated events when a terminal event can censor other non-terminal events but not vice versa. We use a semiparametric frailty model to account for the dependence between correlated survival events and semi-competing risks and adopt penalized partial likelihood (PPL) approach for parameter estimation. In addition, we investigate methods for variable selection in semi-parametric frailty models and propose a double penalized partial likelihood (DPPL) procedure for variable selection of fixed effects in frailty models. We consider two penalty functions, least absolute shrinkage and selection operator (LASSO) and smoothly clipped absolute deviation (SCAD) penalty. The proposed methods are evaluated in simulation studies and illustrated using data from Indianapolis-Ibadan Dementia Project.