Hydraulic fracturing treatments are often used in petroleum and other industries to increase the permeability of rock formations and occur naturally as magmatic intrusions. The resulting fractures, propagating perpendicularly to the minimum in-situ stress, may be widespread due to the significant volumes injected. Because of the difference in the hydrostatic and minimum compressive in-situ stress gradient, they experience a buoyant force. This force elongates the fracture in the direction of gravitational acceleration, and the propagation can become self-sustained without additional fluid being injected. We investigate the impact of these gravitational effects on the emergence, propagation, and arrest of planar three-dimensional hydraulic fractures. We notably show that buoyant hydraulic fractures propagating under continuous fluid releases in impermeable media belong to a family of solutions that depends on a single dimensionless number. This dimensionless number combines the properties of the solid (density, elasticity, fracture resistance), the fracturing fluid (density, viscosity) and the fluid release rate. We also confirm that the emergence of self-sustained buoyant hydraulic fractures in impermeable media is independent of the release history, which governs, however, to the first order how the fracture propagation evolves (the partition between horizontal and vertical growth, ascent rate). We finally study arrest mechanisms for established buoyant hydraulic fractures and show that changes in the fracturing toughness are inefficient in arresting them. Contrarily, stress barriers and fluid mass loss efficiently stop the buoyant propagation of hydraulic fractures.