  We present a sample of 824 solar and late-type stars with X-ray luminosities
and rotation periods. This is used to study the relationship between rotation
and stellar activity and derive a new estimate of the convective turnover time.
From an unbiased subset of this sample the power law slope of the unsaturated
regime, L_X / L_bol = Ro^\beta, is fit as \beta = -2.70 +/- 0.13. This is
inconsistent with the canonical \beta=-2 slope to a confidence of 5 sigma, and
argues for an additional term in the dynamo number equation. From a simple
scaling analysis this implies \Delta\Omega / \Omega = \Omega^0.7, i.e. the
differential rotation of solar-type stars gradually declines as they spin down.
Super-saturation is observed for the fastest rotators in our sample and its
parametric dependencies are explored. Significant correlations are found with
both the corotation radius and the excess polar updraft, the latter theory
providing a stronger dependence and being supported by other observations. We
estimate mass-dependent empirical thresholds for saturation and
super-saturation and map out three regimes of coronal emission. Late F-type
stars are shown never to pass through the saturated regime, passing straight
from super-saturated to unsaturated X-ray emission. The theoretical threshold
for coronal stripping is shown to be significantly different from the empirical
saturation threshold (Ro < 0.13), suggesting it is not responsible. Instead we
suggest that a different dynamo configuration is at work in stars with
saturated coronal emission. This is supported by a correlation between the
empirical saturation threshold and the time when stars transition between
convective and interface sequences in rotational spin-down models.
