Propagation of tensile fractures ubiquitously occurs at depth in the upper earth crust due to fluid pressurization associated either with natural causes (magma overpressure, dehydration reaction in subduction zones, hydrothermal systems etc.) or direct anthropogenic fluid injection (well stimulation for hydrocarbons or geothermal fluids production). The growth of such fluid-driven fractures in a solid under pre-existing compressive stresses is governed by the coupling between lubrication flow inside the growing fracture and linear elastic fracture mechanics of the solid. In nature, the initial stress field becomes more compressive with depth, leading to a buoyancy force generated by the difference between the vertical gradient of the minimum stress (acting perpendicular to the fracture plane) and the fluid weight. Once the fracture grows beyond a critical length scale, this buoyancy force strongly affects its propagation. In particular, the ratio between the energy dissipated by viscous flow and fracture surfaces creation sets the dynamics and elongation of such three-dimensional buoyant fractures. The expectancy and variety of shapes of these fractures will be illustrated in the light of typical material and injection parameters encountered in nature, engineering applications and laboratory experiments. Finally, we will explore how typical variation of in-situ stress and material properties at depth can stop the other-wise self-sustained growth of these fractures.