Motivated by its appearance as a limiting distribution for random and non-random sums of independent random variables, in this paper we develop Stein's method for approximation by the asymmetric Laplace distribution. Our results generalise and offer technical refinements on existing results concerning Stein's method for (symmetric) Laplace approximation. We provide general bounds for asymmetric Laplace approximation in the Kolmogorov and Wasserstein distances, and a smooth Wasserstein distance, that involve a distributional transformation that can be viewed as an asymmetric Laplace analogue of the zero bias transformation. As an application, we derive explicit Kolmogorov, Wasserstein and smooth Wasserstein distance bounds for the asymmetric Laplace approximation of geometric random sums, and complement these results by providing explicit bounds for the asymmetric Laplace approximation of a deterministic sum of random variables with a random normalisation sequence.