In this paper, we review two families for sparsity-aware adaptive filtering. Proportionate-type NLMS filters try to accelerate filter convergence by assigning each filter weight a different gain that depends on its actual value. Sparsity-norm regularized filters penalize the cost function minimized by the filter using sparsity-promoting norms (such as ℓ₀ or ℓ₁) and derive new stochastic gradient descent rules from the regularized cost function. We compare both families of algorithms in terms of computational complexity and studying how well they deal with the convergence vs steady-state error tradeoff. We conclude that sparsity-norm regularized filters are computationally less expensive and can achieve a better tradeoff, making them more attractive in principle. However, selection of the strength of the regularization term seems to be a critical element for the good performance of these filters.