Janus droplets are compound droplets that consist of two adhering drops of different fluids that are suspended in a third fluid. We use the Shan-Chen lattice Boltzmann method for multicomponent mixtures to simulate Janus droplets at rest and in shear. In this simulation model, interfacial tensions are not known a priori from the model parameters and must be determined using numerical experiments. We show that interfacial tensions obtained with the Young-Laplace law are consistent with those measured from the equilibrium geometry. The regimes of adhering, separated, and engulfing droplets were explored. Two different adhesion geometries were considered for two-dimensional simulations of Janus droplets in shear. The first geometry resembles two adhering circles with small overlap. In the second geometry, the two halves are semicircular. For both geometries, the rotation rate of the droplet depends on its orientation. The width of the periodic simulation domain also affects the rotation rate of both droplet types up to an aspect ratio of 6: 1 (width:height). While the droplets with the first geometry oscillated about the middle of the domain, the droplets of the second geometry did not translate while rotating. A four-pole vortex structure inside droplets of the second geometry was found. These simulations of single Janus droplets reveal complex behaviour that implies a rich range of possibilities for the rheology of Janus emulsions. (C) 2014 AIP Publishing LLC.