gamma-rays are a unique probe for extreme events in the universe. Detecting the gamma-rays provides an important opportunity to understand the composition of universe, the evolution of stars, the origin of cosmic rays, etc. gamma-ray astrophysics involves in various frontier scientific issues, and the observed energy spectrum spans over a wide range from a few hundreds of keV to a few hundreds of TeV. Different gamma-ray telescopes are in need for the different scientific goals and spectral bands. In this work, 5 kinds of space- and ground-based gamma-ray observing techniques were summarized including the Coded-aperture telescopes, Compton telescopes, pair-production telescopes, Imaging Atmospheric Cherenkov Telescopes, and Extensive Air Shower Arrays. The progress in gamma-ray astrophysics in the past 70 years, motivated by the observation capability, was reviewed. Great achievements have been made in the high-energy band and very-high-energy band, while because of the limited missions conducted, as well as a lower sensitivity comparing with other bands, discoveries in low- and medium-energy are few, and due to the high observation difficulty, as well as the late start, relevant scientific yields in ultra- and extremely-high energy are limited. Moreover, the future planned missions and capabilities of the gamma-ray telescopes and their possible scientific outputs were discussed. Among these missions, low- and medium-energy space telescopes e-ASTROGAM (enhanced-ASTROGAM), AMEGO (All-sky Medium Energy Gamma-ray Observatory), and very-high-energy ground-based arrays LHAASO (Large High Altitude Air Shower Observatory), CTA (Cherenkov Telescope Array) greatly improve sensitivity than their corresponding last generation, thus expect very likely to further expand our knowledge on the gamma-universe.