This project studies low-rank and sparsity models in real-world problems in data science, such as machine learning. With the development of modern science and technology, we are confronted with massive huge data. How to efficiently understand and use those data is a major problem in data science. Although the data is in high dimension, it is usually determined by a few major factors. So it is reasonable to assume the data is sparse under some basis. How to find the basis as well as its sparse coefficients becomes a hot topic in data science. In this project, we will solve real-world problems in data science using sparsity and low rank models. Through this project, students will have the basic idea on how the sparsity and low rank models are used to solve the real-world problems.