In this thesis, we introduce the basic concepts and properties of submodular functions and its optimization approaches. Then we expore its application in real life setting. To be precise, we explore the application from two aspects:

1. Data subset selection algorithms based on submodular optimization for efficient deep learning model training
2. The application of submodular optimization in social network For the introduction part, we explained the basic background of submodular optimization in detail. And for the empirical analysis part, in order to better present how submodular optimization related methods help develop these two areas, we present detailed experiment results under different settings.