The most prominent intrinsic flame instability is the hydrodynamic, or Darrieus-Landau instability, which arises by virtue of gas expansion that results from the heat released during combustion and is thus ubiquitous to all premixed flames. The Darrieus-Landau instability has many ramifications in premixed combustion; it is responsible for the creation of sharp folds and creases on flame surfaces, for the convex curving of flames propagating in narrow tubes, and for the spontaneous wrinkling that gives outgrowing flames a roughened pebbled-like appearance. The increase in flame surface area creates a commensurate enhancement in fuel consumption or mass-burning rate, which has practical implications in the design of propulsion systems and power generating plants. Although the hydrodynamic instability has been studied extensively, it was not until recently that its effect on turbulent flame propagation began to be examined; it was generally deemed a secondary effect masked by the turbulence and thus largely overlooked. In this paper, we review the main developments that led to the current understanding of the nature of the Darrieus-Landau instability and discuss, in particular, its prominent influence on turbulent flames.