The NASA Science Mission Directorate (SMD) seeks technology for cost-effective high-performance advanced space telescopes for astrophysics and Earth science. Astrophysics applications require large-aperture, lightweight highly reflecting mirrors, deployable large structures, innovative wavefront, structural metrology, and cryogenic optics to enable far-infrared telescopes. A few of the new astrophysics telescopes and their subsystems will require operation at cryogenic temperatures as cold as 4 K. This focus area will consider technologies necessary to enable future telescopes and observatories to collect electromagnetic bands, ranging from X-rays to millimeter waves, and also include gravity waves. New technologies in innovative mirror materials, such as silicon, silicon carbide, and nanolaminates, and innovative structures and deployments are needed on scales ranging from CubeSats to flagship-class missions. Instruments commonly benefit in size and cost from the use of free-form optics and require innovative fabrication and metrology systems. Instrument technologies for high-contrast exoplanet imaging include deformable mirrors and active structures for high-precision wavefront control and stabilization, as well as technologies to accurately deploy and control scattered light with large-scale deployable occulters. Wavelength control and stabilization approaches are critical to the success of high-precision radial velocity measurements. Earth science requires modest apertures in the 2 to 4-meter size category that are cost-effective. Both nanotechnology and wavefront sensing, and control are needed to build telescopes for Earth science.