This thesis considers various aspects of locally covariant quantum field theory (see Brunetti et al., Commun. Math. Phys. 237 (2003), 31-68), a mathematical framework to describe axiomatic quantum field theories in curved spacetimes. Chapter 1 argues that the use of morphisms in this framework can be seen as a model for modal logic. To our knowledge this is the first interpretative description of this aspect of the framework. Chapter 2 gives an exposition of locally covariant quantum field theory which differs from the original in minor details, notably in the new notion of nowhere-classicality and the sharpened time-slice axiom, which puts a restriction on the state space as well as the algebras. Chapter 3 deals with the well-studied example of the free real scalar field and includes an elegant proof of the new general result that the commutation relations together with the Hadamard condition on the two-point distribution of a state completely fix the singularity structure of all n-point distributions. Chapter 4 describes the free Dirac field as a locally covariant quantum field, using a new representation independent approach, demonstrating that the physics is determined entirely by the relations between the adjoint map, charge conjugation and Dirac operator. It also proves the new result that the relative Cauchy evolution is related to the stress-energy-momentum tensor in the same way as for the free scalar field. Chapter 5 studies the Reeh-Schlieder property, both in the general setting and in specific examples. We obtain various interesting results concerning this property in curved spacetimes, most notably by using the idea of spacetime deformation, but some open questions and opportunities for further research remain. We will freely make use of smooth and analytic wave front sets throughout. These concepts are explained in appendix A, using a new and elegant way to generalise results for scalar distributions to Banach space-valued distributions, leading to some new but expected results.