Primitive polynomials over finite fields, and specifically the roots of these polynomials, known as primitive elements, find use in various applications, particularly in the realms of coding theory and cryptography. It is important to not only be able to generate primitive polynomials, but also to be able to guarantee the existence of primitive polynomials satisfying some condition. In this talk, we discuss the problem of finding primitive elements of the form aP+b where P is an element of degree n > 1 over the field GF(q) of q elements, q a prime power, and a and b belong to GF(q). We give a survey of the work done in this area by Davenport and Carlitz, and proceed from there to the more recent results of Cohen, McNay, and Mills. We shall also give attention to a pseudo-random vector generator that one can build using primitive roots of the above form. In addition, we shall talk about primitive polynomials for which certain of its coefficients are prescribed in advance. Cohen resolved the case where the first coefficient is prescribed in advance, and Han settled the case where the first and second coefficients are given in advance for n at least 7, where n is the degree of the primitive polynomial. The remaining cases for the latter problem, namely n = 4, 5, and 6, can be attacked using a sieving technique due to Cohen. We will discuss the work that McNay and Mills have done using the sieve.