The process of coset enumeration has become a significant factor in group theoretical investigations since the advent of modern computing power, but in some respects the process is still not well understood. This thesis investigates some features of coset enumeration, working mainly with the group F(2,7). Chapter 1 describes the characteristics of coset enumeration and algorithms used for it. A worked example of the method is provided. Chapter 2 discusses some features which would be desirable in computer programs for use in investigating the coset enumeration process itself, and reviews the Havas/Alford program which to date best meets the requirements. Chapter 3 deals with the use of coset ammeration in proofs, either in its own right or as a basis for other workings. An example of one attempt to obtain a proof by coset enumeration is given. Chapter 4 reviews techniques designed to reduce the length of coset enumerations and proposes the 'equality list' technique as a way to reduce enumeration length for some groups. Extra insights obtainable using the equality list method are also discussed. Chapter 5 summarises the factors by which the success of different coset enumerations can be compared and proposes an algorithm for making systematic comparisons among enumerations. Chapter 6 reports five coset enumerations, obtained manually by three main methods on the group F(2,7). All these enumerations were shorter than is so far obtainable by machine and one is shorter than other known hand enumerations. The enumerations were compared by applying the process developed in Chapter 5. Chapter 7 presents a shorter proof of the cyclicity of the group F(2,7) than was hitherto available. The proof derives from the workings for one of the coset enumerations described in Chapter 6. There are eight appendices and an annotated bibliography. The appendices contain, inter alia, edited correspondence between well-known coset-enumerators, a guide to the Havas/Alford program, further details on the equality list method and listings of various enumerations.