The recent emergence of various spatially resolved transcriptomic technologies have enabled the study of spatial transcriptomic landscape across a tissue section or within single cells, catalyzing new discoveries in many areas of biology. Despite the fast development of spatial transcriptomic technologies, however, statistical methods for analyzing spatial transcriptomic data are vastly underdeveloped. Analyzing spatial transcriptomic data faces important statistical challenges that arise from the complexities and unique features of these data. Here, we propose to address some of these key statistical challenges in this emerging field through developing a suite of novel statistical methods. Specifically, we will (1) develop Gaussian predictive process models to model the spatial correlation structure in a computational effective way to rapidly identify genes with spatial expression patterns; (2) develop integrative methods to incorporate reference single cell RNA sequencing data along with spatial correlation structure in spatial transcriptomics to enable accurate deconvolution of cell types on the tissue; (3) develop Potts models to perform tissue segmentation and detect tissue regions and microenvironment in a de novo fashion. We will develop, distribute, and support user-friendly open-source software implementing the proposed methods and disseminate them to the scientific community. We will perform rigorous and comprehensive simulations and apply our methods to analyze multiple public spatial transcriptomics data that are collected from different technical platforms and are of different scales. We will also perform an in-depth analysis with supplemental experiments on the spatial transcriptomics data being collected as part of the study of the role of dysregulated stem cell biology in breast cancer disparities in African American women.