Research Doctorate - Doctor of Philosophy (PhD) This thesis focuses on the application of matrix determinants as a means of producing number-theoretic results. Motivated by an investigation of properties of the Riemann zeta function, we examine the growth rate of certain determinants of zeta values. We begin with a generalisation of determinants based on the Hurwitz zeta function, where we describe the arithmetic properties of its denominator and establish an asymptotic bound. We later employ a determinant identity to bound the growth of positive Hankel determinants. Noting the positivity of determinants of Dirichlet series allows us to prove specific bounds on determinants of zeta values in particular, and of Dirichlet series in general. Our results are shown to be the best that can be obtained from our method of bounding, and we conjecture a slight improvement could be obtained from an adjustment to our specific approach. Within the course of this investigation we also consider possible geometric properties which are necessary for the positivity of Hankel determinants, and we examine the role of Hankel determinants in irrationality proofs via their connection with Pade approximation.