This book provides new research in finite fields. Chapter One presents some techniques that rely on a combination of results from graph theory, finite fields, matrix theory, and finite geometry to researchers working in the area of preserver problems. It also gives a brief presentation of this research field to other mathematicians. Chapter Two contains a basic and self-contained introduction to classical coherent state transforms, namely classical wavelet and classical wave-packet transforms, on finite fields. Chapter Three proposes an intrinsic representation of finite mƟ extension as this is a tradition for finite extension fields. Chapter Four reviews mƟ cyclic codes on a mƟ field. Chapter Five discusses two problems of Carlitz and their generalisations.