In directional statistics, the von Mises-Fisher (vMF) distribution has long been a mainstay for inference with data on the unit hypersphere. The performance of statistical inference based on the vMF distribution, however, may suffer when there are significant outliers and noise in the data. Based on an analogy of the median as a robust measure of central tendency and its relationship to the Laplace distribution, we propose the spherical Laplace (SL) distribution, a novel probability measure for modelling directional data. In this paper, we study foundational properties of the distribution such as theoretical results on maximum likelihood estimation and a sampling scheme for probabilistic inference. We derive efficient numerical routines for parameter estimation in the absence of closed-form formula. An application of model-based clustering is considered under the finite mixture model framework. Our numerical methods for parameter estimation and clustering are validated using simulated and real data experiments.