One of the astronomical methods is probably the existence and the development if high end telescopes throughout the years. Telescopes are used by astronomers in viewing the different stars and planets in the universe. It started with a mere observation of the naked eye, then telescopes had evolved to radio and space telescopes.
Telescopes are one of the key instruments for the observation of Universe.

Initially Optical Telescopes were used, then X-ray telescopes and Later on Radio Telescopes were used. They are used till today.

Today, the telescopes can also see in Infrared and Ultraviolet Ranges.

However, the most recent and famous method is to observe the Gravitational Waves. The Laser Interferometer Gravitational Wave Observatory (LIGO) is one such observatory.