  This paper provides an overview of the main ideas driving the bootstrap
algebraic multigrid methodology, including compatible relaxation and algebraic
distances for defining effective coarsening strategies, the least squares
method for computing accurate prolongation operators and the bootstrap cycles
for computing the test vectors that are used in the least squares process. We
review some recent research in the development, analysis and application of
bootstrap algebraic multigrid and point to open problems in these areas.
Results from our previous research as well as some new results for some model
diffusion problems with highly oscillatory diffusion coefficient are presented
to illustrate the basic components of the BAMG algorithm.
