In this talk, I would like to give an overview of recent results by my group on research related to L1 penalized minimization. In particular, 
an approximate Lasso algorithm, BLasso, is proposed and related to L2Boosting; an irrepresentable condition is introduced for Lasso to be 
model selection consistent and the consequence of this condition's relaxation is studied; finally, I will briefly cover a new penalization framework, 
CAP, for grouped and hierarchical selection of predictors, and its fast implementation.