Motivated by the challenges of processing the vast amount of available data, recent research on the ourishing field of high-dimensional statistics is bringing new techniques for modeling and drawing inferences over large amounts of data. Simultaneously, other fields like signal processing and optimization are also producing new methods to deal with large scale problems. More particularly, this work is focused on the theories and methods based on l1-regularization. After a comprehensive review of the l1-norm as tool for finding sparse solutions, we study more deeply the LASSO shrinkage method. In order to show how the LASSO can be used for a wide range of applications, we exhibit a case study on sparse signal processing. Based on this idea, we present the l1 level-slope filter. Experimental results are given for an application on the field of fiber optics communication. For the final part of the thesis, a new estimation method is proposed for high-dimensional models with periodic variance. The main idea of this novel methodology is to combine sparsity, induced by the l1-regularization, with the maximum likelihood criteria. Additionally, this novel methodology is used for building a monthly stochastic model for wind and hydro inow. Simulations and forecasting results for a real case study involving fifty Brazilian renewable power plants are presented.