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This thesis looks into the possibility that X radiation following inner shell ionization by electron impact might be polarized. There has been some speculation on this point: one published conclusion (Cooper and Zare, 1968) is that the polarization must be zero; another conclusion (Mehlhorn, 1968) is that the polarization need not be Jero and can be substantially polarized. By application of the Bethe and Born collision theories (Chapters 4 and 5), it will be shown that both these assertions are wrong: a non-zero polarization can exist, but will be extremely small, even in the region of high impact energies. This work (McFarlane, 1972) has been indirectly confirmed by measurements of the related phenomenon of the angular distribution of Auger electrons following inner shell ionization by electrons (Cleff and Mehlhorn, 1971). By extending the Bethe theory to include relativistic corrections after the manner of Miller (1932) it is shown (Chapter 6) that the polarization approaches its high energy limit only very slowly. The thesis also looks at other anisotropic processes following electron and photon impact. Chapter 7 deals 1 the related problem of Auger electron angular distributions following inner shell photoionization. Chapter 8 postulates a directional correlation between photoelectrons and Auger electrons. Chapter 9 shows that the spin of a photoelectron is correlated with its direction of ejection, if there is significant fine structure interaction in the bound state. An appendix is concerned with the high energy limit of the form of anisotropies, and shows that this limit is more subtle than has been realized. An analytic, compact expression for the line polarization is hence derived and tested successfully against experiment and a more complicated theory.