I came across the book "Cohen-Macaulay Representations" by Graham J. Leuschke and Roger Wiegand, and now I'm wondering if this is an active area of research. If yes, then
what are some of the active problems that people are interested in solving, and what connections does it have to other fields?
Maximal Cohen-Macaulay modules, which seems to be at the center of this book, play a very important role in the theory of non-commutative resolutions singularities. Using MCM modules, Van-den-Bergh gave a very interesting reformulation of Bridgeland's proof of Bondal-Orlov conjecture in dimension 3. See Three dimensional flops and non-commutative rings
For more recent work, you can have a look at any recent paper by Amiot, Donovan, Iyama, Van-den-Bergh-Špenko, Wemyss and their collaborators on the subject.
The classification of CM-finite (meaning there are only finitely many indecomposable CM modules of to isomoprhism) complete local noetherian rings is a wide open problem. The classification is known when the ring is Gorenstein and there is reduces to simple hypersurface singularities of Dynkin type and some exceptional cases. In the non-Gorenstein case, there are just a handful examples known that are CM-finite and it is a big mystery whether there are more. The survey article "Auslander's work on Cohen-Macaulay modules and recent developments" by Yuji Yoshino as well as the book by the same author give a nice overview about this open problem.
