Abstract:
Standard lattice calculations in flavour physics or in studies of hadronic structure are based on the evaluation of matrix elements of local composite operators between hadronic states or the vacuum. In this talk I discuss developments aimed at the computation of long-distance, and hence non-local, contributions to such processes. In particular, I consider the calculation of the $K_L$-$K_S$ mass difference $\Delta m_K=m_{K_L}-m_{K_S}$ and the amplitude for the rare-kaon decay processes $K\to\pi\ell^+\ell^-$, where the lepton $\ell=e$ or $\mu$. Lattice calculations of the long-distance contributions to the indirect $CP$-violating parameter $\epsilon_K$ and to the rare decays $K\to\pi\nu\bar\nu$ are also beginning. Finally I discuss the possibility of including $O(\alpha)$ electromagnetic effects in computations of leptonic and semileptonic decay widths, where the novel feature is the presence of infrared divergences. This implies that contributions to the width from processes with a real photon in the final state must be combined with those with a virtual photon in the amplitude so that the infrared divergences cancel by the Bloch-Nordsieck mechanism. I present a proposed procedure for lattice computations of the $O(\alpha)$ contributions with control of the cancellation of the infrared divergences.

Abstract:
In recent years the precision of lattice calculations has improved hugely, and the results are making a very significant impact in particle physics phenomenology. Indeed there is no alternative general method which can be used in the evaluation of nonperturbative strong interaction effects for a wide variety of physical processes. In this talk I discuss a selection of topics in flavour physics, including \textit{mature} quantities for which lattice calculations have been performed for a long time (e.g. the determination of the $V_{us}$ CKM matrix element and $B_K$), quantities which we are now learning to study (e.g. $K\to\pi\pi$ decays amplitudes and the spectrum and mixing of $\eta-\eta^\prime$ mesons) and important phenomenological quantities for which a large amount of experimental data is available but which we do not yet understand how to approach in lattice simulations (e.g. nonleptonic B-decays). The improvement in precision and the extension of the range of processes which can be studied using lattice QCD has to be continued vigorously if precision flavour physics is to play a complementary role to large $p_\perp$ discovery experiments at the LHC in unravelling the next level of fundamental physics.

Abstract:
I review recent lattice results in kaon physics, particularly in the determination of V_{us} and the B_K parameter of K^0-\bar{K}^0 mixing. I use lattice data to argue for the need of developing SU(2)_L \times SU(2)_R chiral perturbation theory for kaon physics and discuss some recent progress in achieving this. In particular it is shown that for K_{\ell 3} decays at q^2=0 (where q is the momentum transfer between the kaon and the pion), the chiral logarithms can be calculated in spite of the fact that the external pion carries half the energy of the kaon (in the kaon's rest frame), because these logarithms arise from soft internal loops. Future prospects, including applications to K\to\pi\pi decays are discussed. The need to define and exploit renormalization schemes which can simultaneously be implemented numerically in lattice simulations and used in higher-order perturbative calculations is explained.

Abstract:
A review of the status of lattice simulations in particle physics phenomenology is presented. Recent computations of leptonic decay constants of light and heavy mesons, and of the Isgur-Wise function relevant for semi-leptonic decays of $B$-mesons, are discussed in some detail. Calculations of other quantities are briefly outlined. The systematic errors inherent in lattice simulations, and procedures to reduce and control them, are described.

Abstract:
It is shown that the renormalisation constants of two quark operators can be accurately determined (to a precision of a few per-cent using 18 gluon configurations) using Chiral Ward identities. A method for computing renormalisation constants of generic composite operators without the use of lattice perturbation theory is proposed.

Abstract:
The calculation of higher twist (or dimension) corrections to physical quantities using operator product expansions is delicate. If dimensional regularization is used to regulate the ultra-violet divergences then there are ambiguities in the Wilson coefficient functions due to infra-red renormalon singularities. With a hard ultra violet cut-off, such as the inverse lattice spacing $a$, there are no renormalon ambiguities, as a result of cancellations between terms which in finite orders of perturbation theory diverge as inverse powers of $a$, and those which diverge at most logarithmically. In this lecture I review these questions, explaining the steps necessary to obtain predictions for physical quantities from lattice measurements of matrix elements of higher dimensional operators. The ideas are illustrated by considering quantities computed using the heavy quark effective theory beyond leading order in the heavy quark mass.

Abstract:
It is proposed to compute matrix elements for the (unphysical) $K^0\pi^-\to \pi^-$ transition to determine the next-to-leading order low energy constants of the weak chiral Lagrangian. This allows us to evaluate $K\to(\pi\pi)_{I=0}$ decay amplitudes at this level of precision. This approach has several significant advantages over the use of $K\to\pi\pi$ transitions, most notably the elimination of s-channel disconnected diagrams and the use of fewer inversions.

Abstract:
The principal difficulty in deducing weak interaction properties from experimental measurements of $B$-decays lies in controlling the strong interaction effects. In this talk I review the status of theoretical calculations of the amplitudes for exclusive leptonic and semileptonic decays, in the latter case with special emphasis on the extraction of the $V_{cb}$ and $V_{ub}$ matrix elements.

Abstract:
I present a brief introduction to the lattice formulation of quantum field theory, and discuss the use of lattice simulations for studies in particle physics phenomenology. The computation of $f_B$, the decay constant of the $B$-meson, is used as a case study. I also explain the appearance and cancellation of ``renormalons'' in the evaluation of power corrections (higher-twist corrections) in hard scattering and decay processes.

Abstract:
I review the status of theoretical aspects of B-decays. The principal difficulty in interpreting the wealth of experimental data is the control of non-perturbative QCD effects, and the talk is focused on attempts to control these effects. Lattice results for the decay constants, B-$\bar B$ mixing and semileptonic form-factors are summarized. The discrepancy of the theoretical predictions and experimental measurements for the ratio of lifetimes $\tau(\Lambda_b)/\tau(B_0)$ is discussed, as well as the status of the semileptonic branching ratio of the B-meson. The difficulties in making quantitative predictions for exclusive nonleptonic decays are stressed, and some recent approaches to this problem are outlined.