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abstract: 'Two high performance coronagraphic approaches compatible with segmented and obstructed telescope pupils are described. Both concepts use entrance pupil amplitude apodization and a combined phase and amplitude focal plane mask to achieve full coronagraphic extinction of an on-axis point source. While the first concept, named Apodized Pupil Complex Mask Lyot Coronagraph (APCMLC), relies on a transmission mask to perform the pupil apodization, the second concept, named Phase-Induced Amplitude Apodization complex mask coronagraph (PIAACMC), uses beam remapping for lossless apodization. Both concepts theoretically offer complete coronagraphic extinction (infinite contrast) of a point source in monochromatic light, with high throughput and sub-$\lambda$/D inner working angle, regardless of aperture shape. The PIAACMC offers nearly 100% throughput and approaches the fundamental coronagraph performance limit imposed by first principles. The steps toward designing the coronagraphs for arbitrary apertures are described for monochromatic light. Designs for the APCMLC and the higher performance PIAACMC are shown for several monolith and segmented apertures, such as the apertures of the Subaru Telescope, Giant Magellan Telescope (GMT), Thirty Meter Telescope (TMT), the European Extremely Large Telescope (E-ELT) and the Large Binocular Telescope (LBT). Performance in broadband light is also quantified, suggesting that the monochromatic designs are suitable for 