An introductory tutorial on plasma stability theory is presented, focussing on application of the normal mode method to the linearised form of the magnetohydrodynamic, Vlasov and Maxwell equations. Following an illustration of how this method can be applied to a simple mechanical problem, single-fluid stability theory is demonstrated for the case of two spatially separated unmagnetised fluids in the presence of an external force (the Rayleigh–Taylor instability) and a cylindrical magnetised plasma (sausage and kink instabilities). This is followed by a somewhat more heuristic description of two types of instability that play important roles in toroidal plasmas: ballooning modes and tearing modes. The remainder of the article is on two-fluid and kinetic instabilities, starting with the two-stream (Buneman) instability and progressing to its kinetic analogue, the electrostatic bump-in-tail instability, with a detailed discussion of inverse Landau damping, the mechanism that drives it. The article ends with a discussion of how kinetic stability theory can be extended to electromagnetic fluctuations, and the application of this analysis to ion cyclotron emission in magnetised plasmas.