With their jet pointing towards us, blazars are ideal tools to study the physics and structure of extragalactic jets. Their powerful jets are cosmic particle accelerators and are alleged to be one of the production sites of the high-energy neutrinos detected by the IceCube Observatory. Doppler beaming of the jet nonthermal radiation increases blazar brightness, blue-shifts their emission, and shortens their variability time scales, which are observed to range from years down to minutes. This review will focus on blazar flux, spectral, and polarization variability across the electromagnetic spectrum. After introducing blazars and their peculiarities, we will consider the statistical tools that are used to characterize the variability and to reveal correlations and time delays between flux variations at different frequencies. Then we will outline the main observed properties of the blazar multiwavelength behaviour. Interpretation of blazar variability calls into question both intrinsic and extrinsic mechanisms. Shock waves, magnetic reconnection, and turbulence can accelerate particles inside the jet, while jet precession, rotation, and twisting can produce variations in Doppler beaming. Changes in the broad-band spectral energy distribution have commonly been explained by variations in the jet physical parameters in one- or two-zone models. However, microvariability observed at all wavelengths puts strong constraints on the size of the emitting regions, suggesting a multizone emitting jet. Twisting jets have been proposed to explain the long-term multiwavelength variability. They are supported by radio observations of bent or helical jets, and by results of relativistic magnetohydrodynamics simulations of plasma jets. Detection of (quasi)periodic behaviour at all frequencies and on all time scales has been ascribed to orbital motion in black hole binary systems, jet precession, kink instabilities developing inside the jet, or perturbations in the accretion disc. Gravitational microlensing has been suggested to explain blazar behaviour in some cases. Polarization is another important ingredient in blazar variability studies, providing information on the structure and behaviour of the magnetic field in the emission zones. Both the degree and angle of polarization can show strong and fast variability, which is sometimes correlated with flux. Overall, polarimetric observations indicate that turbulence must play an important role in the emitting regions. Recent results obtained by the Imaging X-ray Polarimetric Explorer (IXPE) satellite have revealed some unexpected behaviour favouring a multizone emitting jet model. The interpretation of flux, spectral, and polarization variability within a consistent picture challenges current models of blazar variability and tells us that we may still miss some tiles of the puzzle.
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