A high vacuum, crossed beams apparatus has been constructed which will be used to study Penning ionization and associative ionization. The first experiments to be performed will involve collisions between He(ls2s 21,3S) and various atoms and molecules. The absolute total ionization cross sections, the separate ionization cross sections for all final state positive ions, and the energy distribution of the Penning electrons will be measured. Chapters 1 and 2 of this thesis will describe previous theoretical and experimental work in the field, respectively. Chapter 3 will discuss the design and initial use of the Crossed Beams Apparatus.