In Chapters I to IV of this thesis, the mechanism, originally put forward by Hoyle, Lyttleton and Bondi, by which stars can capture large amounts of interstellar material is examined and extended. The rate of accretion of interstellar material is determined on various assumptions about the nature of this material. The resistive force Is also evaluated under various conditions. The effect of temperature and variation of material density is considered. In Chapter V, a theory of binary star formation is investigated in which the resistive force is considered to remove part of the gravitational energy of a pair of stars and bo leave them gravitationally bound together. It is found that under suitable circumstances such binary star forn&tions can occur, but no estimate has been obtained of the probability of such a formation. In the remaining two chapters, the modifications are considered which have to be applied to the accretion theory when the star is a binary. The dynamical effects of accretion on the binary are considered and estimates are made of the time required for the size of the binary orbit to be appreciably reduced.