This dissertation is concerned with a new formulation of fermionic quantum field theory in classical (electromagnetic or gravitational) backgrounds, which uses methods analogous to those used in conventional multiparticle quantum mechanics. Emphasis is placed on the states of the system, described in terms of Slater determinants, rather than on the field operator, ψ(x). The vacuum state 'at time τ', defined as the Slater determinant of a basis for the span of the negative spectrum of the 'first quantized' Hamiltonian H₁ (τ), provides a concrete realisation of the Dirac Sea. By using the concept of 'radar time', I propose a generalisation of the concept of 'hypersurface of simultaneity', which can be applied to an arbitrarily moving observer in curved spacetime. This is used to provide a consistent particle interpretation for this observer, which depends only on the choice of observer and the background present, not on the choice of coordinates, the choice of gauge (in electromagnetic backgrounds) or the detailed construction of the observer's particle detector. It is also the first definition that does not rely on the spacetime possessing any convenient symmetries. I show that in the cases of a uniformly accelerating observer in flat space (Unruh effect), and a comoving observer in an exponentially inflating universe, my definition reduces to previously accepted definitions. Although this definition is necessarily non-local (no local definition of particle could possibly be consistent with the Unruh effect) I demonstrate with a simple example that this non-locality is only significant on scales of the order of the Compton wavelength λ_c = h/mc of the particle concerned. The general S-matrix element of the theory is derived in terms of time-dependent Bogoliubov coefficients, demonstrating that this follows directly from the definition of inner product between Slater determinants. The process of 'Hermitian extension', inherited directly from conventional multiparticle quantum mechanics, allows second quantized operators to be defined without appealing to a complete set of orthonormal modes, and provides an extremely straightforward derivation of the general expectation value of the theory. Applications of the formalism to pair creation in spatially uniform electric fields, and to the treatment of discrete symmetries, are presented.