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\section*{Abstract}

This thesis investigates various aspects of the relation between the
lattice and continuum formulations of quantum field theories, in
particular QCD.  The aim of this is to gain a better insight into the
theory of QCD, and to be able to relate more accurately the numbers
obtained from lattice simulations to experimental values for physical
quantities.

The first part of this thesis (chapters 1 and 2) gives a general
introduction to quantum field theory, with emphasis on the lattice
formulation of QCD.  The first chapter describes the functional integral
formulation of gauge theories and how it can be used to study these
theories non-perturbatively by discretising the space-time variables.

The second chapter discusses the principles behind the renormalisation
of these theories.  The Ward and Slavnov--Taylor identities that are
preserved non-per\-tur\-ba\-tive\-ly, and can be invoked when
renormalising the theory, are derived.  The final part of this chapter
discusses the renormalisation of composite operators, using both
perturbative and non-perturbative methods.  In particular, it is shown
how the chiral Ward identities can be used to extract renormalisation
constants for the axial and vector currents and the ratio of the scalar
to the pseudoscalar density.

In chapter 3, results for $Z_A$, $Z_V$ and $Z_P/Z_S$ at $\beta=6.2$ are
presented and their effects on calculations of physical quantities like
decay constants are dicussed.

The final chapter investigates the quark--gluon vertex.  The form
factors of the off-shell vertex function, and the symmetries and
Slavnov--Taylor identities that may be used to reduce these form
factors, are discussed.   I then outline a method for extracting the
running coupling from the vertex function.  This also includes a
discussion of the quark and gluon field renormalisation.

Details of computation and results for the vertex function in the Landau
gauge are then presented, and these results are compared with other
determinations of the running coupling and with other more general
studies of the vertex function.
