  Everpresent $\Lambda$ is a cosmological scenario in which the observed
cosmological "constant" $\Lambda$ fluctuates between positive and negative
values with a vanishing mean, and with a magnitude comparable to the critical
density at any epoch. In accord with a longstanding heuristic prediction of
causal set theory, it postulates that $\Lambda$ is a stochastic function of
cosmic time that will vary from one realization of the scenario to another.
Herein, we consider two models of "dark energy" that exhibit these features.
Via Monte Carlo Markov chains, we explore the space of cosmological parameters
and the set of stochastic realizations of these models, finding that
Everpresent $\Lambda$ can fit the current cosmological observations as well as
the $\Lambda$CDM model does. Furthermore, it removes observational tensions
with $\Lambda$CDM, for low redshift measurements of Hubble constant, and the
Baryonic Acoustic Oscillations (BAO) in Lyman-$\alpha$ forest at $z\sim 2-3$.
However, we also find that Everpresent $\Lambda$ does not significantly help
with the growth of ultramassive black holes at high redshift, and the Lithium
problem in Big Bang Nuclesynthesis. Future measurements of "dark energy" at
high redshifts will further test the viability of Everpresent $\Lambda$ as an
alternative to the $\Lambda$CDM cosmology.
