The embedding of discontinuities into finite elements has become a powerful technique for the simulation of fracture in a wide variety of mechanical problems. However, existing formulations still use non-conforming finite elements. In this manuscript, a new conforming formulation is proposed. The main properties of this formulation are as follows: (i) variational consistency; (ii) no limitations on the choice of the parent finite element; (iii) comprehensive kinematics of the discontinuity, including both rigid body motion and stretching; (iv) fully compatible enhanced kinematic field; (v) additional global DOFs located at the discontinuity; (vi) continuity of both jumps and tractions across element boundaries; and (vii) stress locking free. The performance of the proposed formulation is tested by means of academic and structural examples. The numerical results are compared with available experimental results and other numerical approaches, namely the generalized strong discontinuity approach and the generalized FEM/Extended FEM. Copyright © 2012 John Wiley & Sons, Ltd.