Domain walls in strongly coupled gauge theories are discussed. A general mechanism is suggested automatically leading to massless gauge bosons localized on the wall. In one of the models considered, outside the wall the theory is in the nonAbelian confining phase, while inside the wall it is in the Abelian Coulomb phase. Confining property of the non-Abelian theories is a key ingredient of the mechanism which may be of practical use in the context of the dynamic compactification scenarios.In supersymmetric (N = 1) Yang-Mills theories the energy density of the wall can be exactly calculated in the strong coupling regime. This calculation presents a further example of non-trivial physical quantities that can be found exactly by exploiting specific properties of supersymmetry. A key observation is the fact that the wall in this theory is a BPS-saturated state.

In previous papers we have pointed out that there exists a QCD analog of the phenomenological concept of the so-called Fermi motion for the heavy quark inside a hadron. Here we show in a more detailed way how this comes about and we analyze the limitations of this concept. Non-perturbative as well as perturbative aspects are included. We emphasize both the similarities and the differences to the well-known treatment of deep inelastic lepton-nucleon scattering. We derive a model-independent lower bound on the kinetic energy of the heavy quark inside the hadron.

In the last few years considerable progress has been achieved in our understanding of the decays of heavy flavour hadrons. One can now calculate inclusive transition rates in QCD proper through an expansion in inverse powers of the heavy flavour quark mass without recourse to phenomenological assumptions. The non-perturbative contributions are treated systematically in this way; they are found to produce corrections of order a few percent in beauty decays, i.e. typically somewhat smaller than the perturbative corrections. One finds, among other things: (a) The lifetime of B − mesons is predicted to be longer than that of B 0 mesons by several percent. (b) The QCD prediction for the semileptonic branching ratio of B mesons appears to exceed present experimental values. We discuss the implications of this discrepancy. The phenomenological engineering that has been developed for the description of exclusive two-body modes of B mesons has reached a mature stage and awaits more precise and detailed experimental tests. First steps towards a genuine QCD treatment of these modes are being made. 'Anyone who keeps the ability to see beauty never grows old'Franz Kafka 1 To appear in the second edition of the book 'B Decays', S.Stone (ed.), World Scientific 2 Permanent address

We discuss a new class of spectral problems discovered recently which occupies an intermediate position between the exactly-solvable problems (like the famous harmonic oscillator) and all others. The problems belonging to this class are distinguished by the fact that an (arbitrary) part of the eigenvalues and eigenfunctions can be found algebraically, but not the whole spectrum. The reason explaining the existence of the quasi-exactly-solvable problems is a hidden dynamical symmetry present in the Hamiltonian. For one-dimensional motion, this hidden symmetry is SU(2). The simplest one-dimensional system admitting algebraization for a part of the spectrum is the anharmonic oscillator with the x6 anharmonicity and a relation between the coefficients in front of x2 and x6. We review also more complicated cases with the emphasis on pedagogical aspects. The groups SU (2)× SU (2), SO(3) and SU(3) generate two-dimensional problems with the partial algebraization of the spectrum. Typically we get Schrödinger-type equations in curved space. An intriguing relation between the algebraic structure of the Hamiltonian and the geometry of the space emerges. Another interesting development is the use of the graded algebras which allow one to construct multi-component quasi-exactly-solvable Hamiltonians.