New families of three-dimensional nonlinear travelling waves are discovered in pipe flow. In contrast to known waves (Faisst & Eckhardt Phys. Rev. Lett. 91, 224502 (2003), Wedin & Kerswell, J. Fluid Mech. 508, 333 (2004)), they possess no rotational symmetry and exist at much lower Reynolds numbers. Particularly striking is an `asymmetric mode' which has one slow streak sandwiched between two fast streaks located preferentially to one side of the pipe. This family originates in a pitchfork bifurcation from a mirror-symmetric travelling wave which can be traced down to a Reynolds number of 773. Helical and non-helical rotating waves are also found emphasizing the richness of phase space even at these very low Reynolds numbers. The delay in Reynolds number from when the laminar state ceases to be a global attractor to turbulent transition is then even larger than previously thought.