  I will review recent developments in matrix deflation methods, by Ronald
Morgan/Walter Wilcox, Andreas Stathopoulos/Konstantinos Orginos, and Martin
L\"uscher, with application to lattice QCD fermion inversion. I will begin with
a short review of deflation-related work in the field. The Morgan/Wilcox
algorithms using GMRES and BiCGStab for deflation will be described. Typical
results for quenched (Wilson and twisted mass) and dynamical configurations
(CP-PACS and ETMC) will be displayed and discussed. I will outline how
inclusion of multiple right-hand sides and multiple shifts can be accommodated
within versions of these methods. I will also review deflation methods based on
Conjugate Gradient introduced by Stathopoulos and Orginos for hermitian systems
with multiple right-hand sides as well as L\"uscher's non-hermitian domain
decomposition algorithm using Generalized Conjugate Residual.}
