The Low EMittance Muon Accelerator (LEMMA) concept uses a positron-driven muon source, providing an attractive path to very high energy lepton colliders with improved particle backgrounds. The recent developments of the LEMMA overall layout will be described, together with its main challenges. The positron beam is stored in a ring with high energy acceptance and low emittance and then extracted into a multi-IP beamline to produce muon pairs at threshold on the target’s electrons. The muon target is being optimized in terms of material and thickness. The produced muon beam is being characterized from the source through the accumulator rings up to the accelerating sections, with the final goal of a start-to-end simulation for the muon beams up to the collider.