Light is a wave. This was the verdict by 1850 — Young, Fresnel, Maxwell all agreed. Then in 1905, Einstein proposed that light comes in discrete chunks, called quanta (later photons), with energy E = hν proportional to frequency. He was explaining the photoelectric effect: energy of the ejected electrons depends on the light's frequency, not its intensity. The Nobel Committee gave Einstein the 1921 Prize not for relativity but for the photoelectric paper. By the 1920s, Louis de Broglie proposed the converse: if waves can behave like particles, then particles can behave like waves. Electrons, protons, atoms all turned out to interfere with themselves. Wave-particle duality — every quantum object is both — is the central interpretive puzzle of quantum mechanics.
Light exhibits wave behavior: interference patterns in the double-slit experiment, diffraction, polarization. Light also exhibits particle behavior: the photoelectric effect (Einstein, 1905) shows discrete energy quanta; Compton scattering (1923) shows discrete momentum exchange; individual photons can be detected one at a time. The two pictures resolve once you accept that light is described by a quantum field whose excitations behave as wave-like quanta — neither classical wave nor classical particle, but a third thing. Particles exhibit wave behavior too. Electron diffraction (Davisson-Germer, 1927) confirmed de Broglie's hypothesis: an electron of momentum p has wavelength λ = h/p. Neutron diffraction and atom interferometry extended the result to heavier objects; molecular interferometry with C₆₀ buckminsterfullerenes (1999) and biological molecules (2019) showed quantum interference for macroscopic-scale objects. The double-slit experiment with single particles: send electrons one at a time; over many runs an interference pattern builds up, even though each arrived as a localized hit. Each particle interferes with itself. Try to detect which slit a particle went through and the pattern disappears — the particle is now localized. The act of measurement changes the outcome. Interpretive schools — Copenhagen, Many-Worlds, Bohmian mechanics, QBism — make the same predictions but disagree about what is actually happening. The mathematics is settled: quantum systems are described by wavefunctions evolving via the Schrödinger equation; measurement extracts probabilistic outcomes via the Born rule.
Electron microscopes use the wave nature of electrons to image at sub-atomic resolution, far better than any optical microscope. X-ray crystallography — which revealed the structures of DNA, haemoglobin, and almost all of structural biology — exploits X-rays diffracting off crystalline atom arrays. Quantum computers exploit quantum superposition and interference of qubit states — wave-particle duality at engineering level. Atomic clocks (the cesium clocks that define the SI second) use atomic transitions between quantized energy levels. The principle that quantum objects are neither waves nor particles but a third thing with elements of both is now the standard premise of all quantum technology.