Abstract This paper presents a formulation for frictionless beam-to-beam contact using the mortar method. The beams are modelled using the geometrically exact theory. A similar approach has been proposed very recently, with respect to which we offer a formulation based on a Lagrange-multiplier method and a simpler algorithm to cover the static interaction within the contact zone and analyse the performance of the method for different orders of interpolation for the Lagrange multiplier and in the presence of self-contact. Appropriate contact kinematics is developed from which the residual vector and the tangent stiffness matrix are obtained from a suitable contact potential and its variation and consistent linearisation for implementation in the finite element method. The algorithm describing the fulfilment of the contact kinematics is described in detail. The mortar method is found out to be suitable for modelling beam-to-beam contact and self-contact. The geometrically exact beam theory assumes full rigidity of the cross-sections and as such is naturally prone to higher oscillations in the contact force near the boundaries of the contact zone. For sufficiently small load steps, however, a stable solution is obtained, making it appropriate for future research.