Joint work with Roger Wiegand.

**Abstract:**

The Brauer-Thrall Conjectures, now theorems, were originally stated for finitely generated modules over a finite-dimensional *k*-algebra. They say, roughly speaking, that infinite representation type implies the existence of lots of indecomposable modules of arbitrarily large *k*-dimension. These conjectures have natural interpretations in the context of maximal Cohen-Macaulay modules over Cohen-Macaulay local rings. This is a survey of progress on these transplanted conjectures.

**Notes:**

Much of this material appears in our book, but here it is collected in one place, and explained a bit more clearly. The paper also lists some open questions.

**Bibtex code:**

@incollection {MR3051386, AUTHOR = {Leuschke, Graham J. and Wiegand, Roger}, TITLE = {Brauer-{T}hrall theory for maximal {C}ohen-{M}acaulay modules}, BOOKTITLE = {Commutative algebra}, PAGES = {577--592}, PUBLISHER = {Springer}, ADDRESS = {New York}, YEAR = {2013}, MRCLASS = {13C14}, MRNUMBER = {3051386}, DOI = {10.1007/978-1-4614-5292-8_18}, URL = {http://dx.doi.org/10.1007/978-1-4614-5292-8_18}, }