In　this　paper,　we　provide　some　gentle　introductions　to　the　recent　advance　in　augmented　Lagrangian　methods　for　solving　large-scale　convex　matrix　optimization　problems　(cMOP).　Specifically,　we　reviewed　two　types　of　sufficient　conditions　for　ensuring　the　quadratic　growth　conditions　of　a　class　of　constrained　convex　matrix　optimization　problems　regularized　by　nonsmooth　spectral　functions.　Under　a　mild　quadratic　growth　condition　on　the　dual　of　cMOP,　we　further　discussed　the　R-superlinear　convergence　of　the　Karush–Kuhn–Tucker　(KKT)　residuals　of　the　sequence　generated　by　the　augmented　Lagrangian　methods　(ALM)　for　solving　convex　matrix　optimization　problems.　Implementation　details　of　the　ALM　for　solving　core　convex　matrix　optimization　problems　are　also　provided.　©　2021,　Operations　Research　Society　of　China,　Periodicals　Agency　of　Shanghai　University,　Science　Press,　and　Springer-Verlag　GmbH　Germany,　part　of　Springer　Nature.