Status of lattice calculations of hadron matrix elements along with CP violation inB and inK systems is reviewed. Lattice has provided useful input which, in conjunction with experimental data, leads to the conclusion that CP-odd phase in the CKM matrix plays the dominant role in the observed asymmetry inB → ψK_s. It is now quite likely that any beyond the SM, CP-odd, phase will cause only small deviations in B-physics. Search for the effects of the new phase(s) will consequently require very large data samples as well as very precise theoretical predictions. Clean determination ofall the angles of the unitarity triangle therefore becomes essential. In this regardB → KD⁰ processes play a unique role. RegardingK-decays, remarkable progress made by theory with regard to maintenance of chiral symmetry on the lattice is briefly discussed. First application already provide quantitative information onB_K and the ΔI = 1/2 rule. In the lattice calculation, the enhancement in Re A₀ appears to arise solely from tree operators, esp. Q₂; penguin contribution toRe A₀ appears to be very small. However, improved calculations are necessary for ε’/ε as the contributions of QCD penguins and electroweak penguins largely seem to cancel. There are good reasons, though, to believe that these cancellations will not survive improvements that are now underway. Importance of determining the unitarity triangle purely fromK-decays is also emphasized.

The importance of a super-B factory in the search for new physics, in particular, due to CP-odd phase(s) from physics beyond the standard model is surveyed. The first point to emphasize is that we now know how to directly measure all three angles of the unitarity triangle very cleanly, i.e. without theoretical assumptions with irreducible theory error ≲1%. However, this requires much more luminosity than is currently available atB-factories. Direct searches via penguin-dominated hadronic modes as well as radiative, pair-leptonic and semi-leptonic decays are also discussed. Null tests of the SM are stressed as these will play a crucial role especially if the effects of BSM phase(s) onB-physics are small.

Although the CKM-paradigm seems to work to an accuracy of about 15–20%, we emphasize that there are by now several indications that suggest the need for a beyond the Standard Model CP-odd phase. The value of $\sin 2\beta$ measured via the gold-plated (tree) mode, $B \rightarrow \psi K_{\text{s}}$ is smaller than the value deduced by using improved lattice matrix elements. The value of $\sin 2\beta$ measured via `penguin-dominated` (loop) decays tends to be even smaller still. There is also a rather large difference between the direct CP asymmetries in $\bar{B}^{0} \rightarrow K^{−} \pi^{+}$ and $B^{−} \rightarrow K^{−} \pi^{0}$ that is rather difficult to understand. More recently, CDF and D0 are finding about a $2\sigma$ signal in CP asymmetry in the corresponding gold-plated mode $B_{\text{s}} \rightarrow \psi \phi$. If true, this would be consistent with the indications of new CP-phase in penguin $b \rightarrow s$ transitions seen at 𝐵-factories. After describing these possible signs of trouble for the SM-CKM paradigm, we give a brief discussion of some of the BSM scenarios that could be the underlying cause of these deviations. In particular, we find that the data are quite suggestive of a fourth family with $m_{t}^{ˊ} in the range of 400–600 GeV as perhaps the simplest BSM candidate which `naturally' explains the data.