  We present results for the \Delta S=2 matrix elements which are required to
study neutral kaon mixing in the standard model (SM) and beyond (BSM). We also
provide leading chiral order results for the matrix elements of the electroweak
penguin operators which give the dominant \Delta I=3/2 contribution to direct
CP violation in K->\pi\pi decays. Our calculations were performed with
Neuberger fermions on two sets of quenched Wilson gauge configurations at
inverse lattice spacings of approximately 2.2 GeV and 1.5 GeV. All
renormalizations were implemented non-perturbatively in the RI/MOM scheme,
where we accounted for sub-leading operator product expansion corrections and
discretization errors. We find ratios of non-SM to SM matrix elements which are
roughly twice as large as in the only other dedicated lattice study of these
amplitudes. On the other hand, our results for the electroweak penguin matrix
elements are in good agreement with two recent domain-wall fermion
calculations. As a by-product of our study, we determine the strange quark
mass. Our main results are summarized and discussed in Sec. VII. Within our
statistics, we find no evidence for scaling violations.
