Several designs for high-energy Lepton Colliders serving as Higgs factories but extendable to higher energies up to the TeV range are under discussion. The most mature design is the International Linear Collider (ILC), but also the Compact Linear Collider (CLIC) as well as the new concept of a Hybrid Asymmetric Linear Higgs Factory (HALHF) have a large physics potential. The first energy stage with $\sqrt{s}=250$~GeV requires high luminosity and polarized beams and imposes an effort for all positron source designs at high-energy colliders. In the baseline design of the ILC, an undulator-based source is foreseen for the positron source in order to match the physics requirements. In this contribution an overview is given about the undulator-based source, the target tests, the rotating target wheel design, as well as the pulsed solenoid and the new technology development of plasma lenses as optic matching devices.