It is well known that a smooth $2n$-dimensional symplectic manifold $N$ is equipped with a closed, nondegenerate differential 2-form $w$. An $n$-dimensional submanifold $L$ of $N$ is called Lagrangian if the restriction of $w$ to $L$ is zero. Lagrangian submanifolds and their deformations have important applications in symplectic geometry and mathematical physics. In particular, they play a role in establishing the correspondence between 'Calabi-Yau mirror pairs' in string theory via the Fukaya category. In this talk, we first study the deformations of (special) Lagrangian submanifolds and then extend the theory to 'Lagrangian type' submanifolds inside $G_2$ manifolds.