In the first part of this thesis, we survey and summarize some known re- sults in Lagrangian and special Lagrangian geometry. In particular we fo- cus on the structure of the moduli space of special Lagrangian subamanifolds studied by Hitchin.
In the second part, we focus on the theory of Lagrangian mean curvature flow, especially the Lagrangian self-similar solutions constructed by Joyce- Lee-Tsui. At the end we describe an unsolved conjecture given by Thomas and Yau.
