The human tongue is composed mainly of skeletal muscle tissue and has a complex architecture. Its anatomy is characterised by interweaving yet distinct muscle groups. It is a significant contributor to the phenomenon of obstructive sleep apnoea syndrome. A realistic model of the tongue and computational simulations are important in areas such as linguistics and speech therapy. The aim of this work is to report on the construction of a geometric and constitutive model of the human tongue and to demonstrate its use in computational simulations for obstructive sleep apnoea syndrome research. The geometry of the tongue and each muscle group of the tongue, including muscle fibre orientations, are captured from the Visible Human Project dataset. The fully linear muscle model is based on the Hill three-element model that represents the constituent parts of muscle fibres. The mechanics of the model are limited to quasi-static, small-strain, linear-elastic behaviour. The main focus of this work is on the material directionality and muscle activation. The transversely isotropic behaviour of the muscle tissue is accounted for, as well as the influence of muscle activation. The behaviour of the model is illustrated in a number of benchmark tests and for the case of a subject in the supine position. Copyright © 2013 John Wiley & Sons, Ltd.