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We survey our understanding of classical novae—nonterminal, thermonuclear eruptions on the surfaces of white dwarfs in binary systems. The recent and unexpected discovery of GeV gamma rays from Galactic novae has highlighted the complexity of novae and their value as laboratories for studying shocks and particle acceleration. We review half a century of nova literature through this new lens, and conclude the following: The basics of the thermonuclear runaway theory of novae are confirmed by observations. The white dwarf sustains surface nuclear burning for some time after runaway, and until recently, it was commonly believed that radiation from this nuclear burning solely determines the nova's bolometric luminosity. The processes by which novae eject material from the binary system remain poorly understood. Mass loss from novae is complex (sometimes fluctuating in rate, velocity, and morphology) and often prolonged in time over weeks, months, or years. The complexity of the mass ejection leads to gamma-ray-producing shocks internal to the nova ejecta. When gamma rays are detected (around optical maximum), the shocks are deeply embedded and the surrounding gas is very dense. Observations of correlated optical and gamma-ray light curves confirm that the shocks are radiative and contribute significantly to the bolometric luminosity of novae. Novae are therefore the closest and most common interaction-powered transients. [ABSTRACT FROM AUTHOR]