Many semidefinite programs (SDPs) arising in practical applications have useful structural properties that can be exploited at the algorithmic level. In this chapter, we review two decomposition frameworks for large-scale SDPs characterized by either chordal aggregate sparsity or partial orthogonality. Chordal aggregate sparsity allows one to decompose the positive semidefinite matrix variable in the SDP, while partial orthogonality enables the decomposition of the affine constraints. The decomposition frameworks are particularly suitable for the application of first-order algorithms. We describe how the decomposition strategies enable one to speed up the iterations of a first-order algorithm, based on the alternating direction method of multipliers, for the solution of the homogeneous self-dual embedding of a primal-dual pair of SDPs. Precisely, we give an overview of two structure-exploiting algorithms for semidefinite programming, which have been implemented in the open-source MATLAB solver CDCS. Numerical experiments on a range of large-scale SDPs demonstrate that the decomposition methods described in this chapter promise significant computational gains.