This book concerns existence of primitive polynomials over finite fields with an arbitrarily prescribed coefficient. It completes the proof of a fundamental conjecture of Tom Hansen and Gary L. Mullen (1992) which asserts that, with some explicable general exceptions, there always exists a primitive polynomial of any degree over any finite field with an arbitrary coefficient prescribed. Here, the last remaining cases of the conjecture are proven efficiently, in a self-contained way and with very little computation. This is achieved by separately considering the polynomials with second, third or fourth coefficient prescribed, and in each case developing methods involving the use of character sums and sieving techniques. When the characteristic of the field is 2 or 3, p-adic analysis is used. The book also researches the existence of primitive polynomials with two coefficients prescribed (the constant term and any other coefficient).