  We provide a comprehensive discussion of the Everpresent $\Lambda$
cosmological model arising from fundamental principles in causal set theory and
unimodular gravity. In this framework the value of the cosmological constant
($\Lambda$) fluctuates, in magnitude and in sign, over cosmic history. At each
epoch, $\Lambda$ stays statistically close to the inverse square root of the
spacetime volume. Since the latter is of the order of $H^2$ today, this
provides a way out of the cosmological constant puzzle without fine tuning. Our
discussion includes a review of what is known about the topic as well as new
motivations and insights supplementing the original arguments. We also study
features of a phenomenological implementation of this model, and investigate
the statistics of simulations based on it. Our results show that while the
observed values of $H_0$ and $\Omega_\Lambda^0$ are not typical outcomes of the
model, they can be achieved through a modest number of simulations. We also
confirm some expected features of $\Lambda$ based on this model, such as the
fact that it stays statistically close to the value of the total ambient energy
density (be it matter or radiation dominated), and that it is likely to change
sign roughly every Hubble timescale.
