This thesis consists of four research papers focusing on estimation and inference in missing data. In the first paper (Chapter 2), an approximate Bayesian approach is developed to handle unit nonresponse with parametric model assumptions on the response probability, but without model assumptions for the outcome variable. The proposed Bayesian method is also extended to incorporate the auxiliary information from full sample. In second paper (Chapter 3), a new Bayesian method using the Spike-and-Slab prior is proposed to handle the sparse propensity score estimation. The proposed method is not based on any model assumption on the outcome variable and is computationally efficient. In third paper (Chapter 4), we develop a robust semiparametric method based on the profile likelihood obtained from semiparametric response model. The proposed method uses the observed regression model and the semiparametric response model to achieve robustness. An efficient algorithm using fractional imputation is developed. The bootstrap testing procedure is also proposed to test ignorability assumption. In last paper (Chapter 5), we propose a novel semiparametric fractional imputation method using Gaussian mixture model for handling multivariate missingness. The proposed method is computationally efficient and leads to robust estimation. The proposed method is further extended to incorporate the categorical auxiliary information. Asymptotic properties are developed for each proposed methods. Both simulation studies and real data applications are conducted to check the performance of the proposed methods in this thesis.