Michelson–Morley experiment
The first breakthrough was caused by the results of the so-called Michelson–Morley experiment. In 1887 the two American physicists, Michelson and Morley, were trying to detect the theorized ether, a medium through which electromagnetic waves (predicted by Maxwell's equations) were supposed to travel. Nowadays, we know that light does not in fact need a medium to propagate through, unlike sound waves requiring air or waves on a pond requiring water.
Michelson and Morley presumed that ether is not static, but is dragged along by the rotation of the Earth for instance. Thus they measured the speed of light in perpendicular directions expecting varying velocities based on the relative motion of the medium. Nevertheless, they measured a constant speed of light every time, as predicted by Maxwell, at about $c \approx 3×10^8 m/s$.
This was however in conflict with one of the building blocks of physics, Newton’s laws of motion. Newton stated that the speed of light, as any other velocity, should be relative and dependent on the reference frame. Subsequently, this contradiction was resolved by Albert Einstein in 1905 with the publication of the Special theory of relativity in favour of the more substantiated Maxwell’s equations, as Newton’s laws of motion turned out to be merely an approximation applicable only at low speeds.
Blackbody radiation
A second open question in physics of the 19th century was the discrepancy between theory and observation of black-body radiation, which is also known as the Ultraviolet catastrophe. The relationship between the spectral distribution of thermal radiation spontaneously emitted by an ideal colourless and nonreflective object and its temperature was well-known at the time from experiments. An established classical theory, however, predicted that as the wavelength of the emitted light decreased (corresponding to ultraviolet light) its intensity would increase without any limit.
Ultraviolet catastrophe
In 1900, German physicist Max Planck derived an equation (Planck’s law) that predicted the observed spectrum perfectly, by assuming that there is a certain minimal amount of energy black-body can emit, i.e., the radiation is emitted in packets not continuously.
Photoelectric effect
In 1905, Albert Einstein developed Planck’s work further and quantized all light into discrete packets called photons to explain the last unanswered question of the physics of the times, the photoelectric effect. Photoelectric effect can be observed when a source of light is pointed at a material, let’s say a metal plate, which causes the material to emit electrons.
When the intensity of the light is increased, the rate of electron emission remains unchanged. Whereas when the frequency of the light is increased (corresponding to a decreasing wavelength), the more electrons are released.
Classical theory could not explain this phenomenon, until Albert Einstein theorized, that light in this scenario behaves as a particle rather than a continuous wave. The higher frequency light was then equivalent to a higher number of separate photons hitting the metal plate and higher energy.
Photoelectric effect