Parts I and II deal with the theory of crystal growth, parts III and IV with the form (on the atomic scale) of a crystal surface in equilibrium with the vapour. In part I we calculate the rate of advance of monomolecular steps (i.e. the edges of incomplete monomolecular layers of the crystal) as a function of supersaturation in the vapour and the mean concentration of kinks in the steps. We show that in most cases of growth from the vapour the rate of advance of steps will be independent of their crystallographic orientation, so that a growing closed step will be circular. We also find the rate of advance for parallel sequences of steps, and the dependence of rate of advance upon the curvature of the step. In part II we find the resulting rate of growth and the steepness of the growth cones or growth pyramids when the persistence of steps is due to the presence of dislocations. The cases in which several or many dislocations are involved are analysed in some detail; it is shown that they will commonly differ little from the case of a single dislocation. The rate of growth of a surface containing dislocations is shown to be proportional to the square of the supersaturation for low values and to the first power for high values of the latter. Volmer & Schultze's (1931) observations on the rate of growth of iodine crystals from the vapour can be explained in this way. The application of the same ideas to growth of crystals from solution is briefly discussed. Part III deals with the equilibrium structure of steps, especially the statistics of kinks in steps, as dependent on temperature, binding energy parameters, and crystallographic orientation. The shape and size of a two-dimensional nucleus (i.e. an 'island' of new monolayer of crystal on a completed layer) in unstable equilibrium with a given supersaturation at a given temperature is obtained, whence a corrected activation energy for two-dimensional nucleation is evaluated. At moderately low supersaturations this is so large that a perfect crystal would have no observable growth rate. For a crystal face containing two screw dislocations of opposite sense, joined by a step, the activation energy is still very large when their distance apart is less than the diameter of the corresponding critical nucleus; but for any greater separation it is zero. Part IV treats as a 'co-operative phenomenon' the temperature dependence of the structure of the surface of a perfect crystal, free from steps at absolute zero. It is shown that such a surface remains practically flat (save for single adsorbed molecules and vacant surface sites) until a transition temperature is reached, at which the roughness of the surface increases very rapidly ('surface melting'). Assuming that the molecules in the surface are all in one or other of two levels, the results of Onsager (1944) for two-dimensional ferromagnets can be applied with little change. The transition temperature is of the order of, or higher than, the melting-point for crystal faces with nearest neighbour interactions in both directions (e.g. (100) faces of simple cubic or (111) or (100) faces of face-centred cubic crystals). When the interactions are of second nearest neighbour type in one direction (e.g. (110) faces of s.c. or f.c.c. crystals), the transition temperature is lower and corresponds to a surface melting of second nearest neighbour bonds. The error introduced by the assumed restriction to two available levels is investigated by a generalization of Bethe's method (1935) to larger numbers of levels. This method gives an anomalous result for the two-level problem. The calculated transition temperature decreases substantially on going from two to three levels, but remains practically the same for larger numbers.