This paper presents an isogeometric (IGA) solver for time-dependent, incompressible magnetohydrodynamics (MHD). In this paper, a combination of inf-sup stable mixed discretisations is considered to discretise the hydrodynamic pair (i.e. velocity and pressure) and magnetic pair (i.e. magnetic field and magnetic pressure). The one step θ-method is used for the temporal discretisation. Manufactured and analytical solutions are considered to assess convergence, accuracy and performance of the IGA solver. Moreover, benchmark problems involving a MHD flow over an obstacle and MHD flow under non-uniform magnetic field are considered to examine the effect of the magnetic field on the flow behaviour.