The analysis of univariate and multivariate longitudinal data (U/MLD) with censored and missing response has inspired considerable interest in the statistical community recently. In the case of MLD, estimating the contemporaneous correlation coefficient is of particular interest to applied researchers. In this dissertation, mixed model methodologies are investigated to analyze U/MLD with censored and missing response while accounting for complex features of longitudinal data such as serial correlation and/or heteroscedasticity of the measurement error, and multiple level of nesting in the random effects. We propose a computationally feasible expectation maximization (EM) algorithm for maximum likelihood (ML) and restricted maximum likelihood (REML) estimation for univariate and bivariate linear mixed effects models (U/BLMEMs) with censored and missing response. We implement our proposed methodology in an R function lmecm, which computes ML and REML for U/BLMEMs with censored and missing response and fits a wider class of models than existing computational tools. The performance of the proposed methods is evaluated through extensive simulation studies and analysis of real data from environmental and biomedical studies.