A set of lines passing through the origin in Euclidean space is called equiangular if the angle between any two lines is the same. The question of finding the maximum number of such lines, N(d) in any dimension d is an extensively studied problem. Closely related, is the problem of finding the maximum number of lines, N_α(d), such that the common angle between the lines is arccosα. In recent years, many progress has been made on this problem. We review some of these breakthrough results and the techniques they use to approach this problem. The first main result is a linear upper bound on N_α(d) which is found using a completely novel approach with respect to techniques used in previous works. Another main result that we discuss solves the problem of finding N_α(d) for high enough dimensions. Some classic results from some of the first studies on equiangular lines are also discussed. Finally, some suggestions are given for possible further research.

This thesis focusses on good ultrafilters. Firstly ultrafilters are introduced and the definition of a good ultrafilter is given. Next the proof that good ultrafilters exists is discussed. Lastly it is shown that good ultrafilters make ultraproducts saturated.