The batch least-absolute shrinkage and selection operator (Lasso) has well-documented merits for estimating sparse signals of interest emerging in various applications, where observations adhere to parsimonious linear regression models. To cope with linearly growing complexity and memory requirements that batch Lasso estimators face when processing observations sequentially, the present paper develops a recursive Lasso algorithm that can also track slowlyvarying sparse signals of interest. Performance analysis reveals that recursive Lasso can either estimate consistently the sparse signal's support or its nonzero entries, but not both. This motivates the development of a weighted version of the recursive Lasso scheme with weights obtained from the recursive least-squares (RLS) algorithm. The resultant RLS-weighted Lasso algorithm provably estimates sparse signals consistently. Simulated tests compare competing alternatives and corroborate the performance of the novel algorithms in estimating time-invariant and tracking slow-varying signals under sparsity constraints.