This chapter introduces the concepts of Lie groups, Lie algebras, and their representations, which are crucial in physics for encoding symmetries. A Lie group is a continuous group that is also a smooth manifold, with operations of inversion and multiplication being smooth. The tangent space at the identity of a Lie group forms a Lie algebra, which encodes many properties of the group but is easier to work with due to its linear structure. In quantum mechanics, symmetry is often represented through unitary or projective unitary representations of the symmetry group on the Hilbert space. These representations are homomorphisms from the group to the group of unitary operators on the Hilbert space. For example, the Hamiltonian operator of a quantum system is invariant under rotations if it commutes with the representation of the rotation group and its associated Lie algebra operators. The chapter focuses on matrix Lie groups, which are closed subgroups of invertible matrices. Each matrix Lie group \( G \) has a corresponding Lie algebra \( \mathfrak{g} \), which is a real subspace of the space of all \( n \times n \) matrices. The Lie algebra comes with a bracket operation that encodes information about the group. Unitary representations of \( G \) are continuous homomorphisms into the group of unitary operators on a Hilbert space. Each representation of \( G \) gives rise to a representation of its Lie algebra, but not every representation of the Lie algebra comes from a representation of the group, except when \( G \) is simply connected.This chapter introduces the concepts of Lie groups, Lie algebras, and their representations, which are crucial in physics for encoding symmetries. A Lie group is a continuous group that is also a smooth manifold, with operations of inversion and multiplication being smooth. The tangent space at the identity of a Lie group forms a Lie algebra, which encodes many properties of the group but is easier to work with due to its linear structure. In quantum mechanics, symmetry is often represented through unitary or projective unitary representations of the symmetry group on the Hilbert space. These representations are homomorphisms from the group to the group of unitary operators on the Hilbert space. For example, the Hamiltonian operator of a quantum system is invariant under rotations if it commutes with the representation of the rotation group and its associated Lie algebra operators. The chapter focuses on matrix Lie groups, which are closed subgroups of invertible matrices. Each matrix Lie group \( G \) has a corresponding Lie algebra \( \mathfrak{g} \), which is a real subspace of the space of all \( n \times n \) matrices. The Lie algebra comes with a bracket operation that encodes information about the group. Unitary representations of \( G \) are continuous homomorphisms into the group of unitary operators on a Hilbert space. Each representation of \( G \) gives rise to a representation of its Lie algebra, but not every representation of the Lie algebra comes from a representation of the group, except when \( G \) is simply connected.