  Despite the many efforts, our theoretical understanding of the ultimate
nature of the dark energy component of the universe still lags well behind the
astounding experimental evidence achieved from the increasingly sophisticated
observational tools at our disposal. While the canonical possibility is a
strict cosmological constant, or rigid vacuum energy density
$\rho_{\Lambda}=$const., the exceeding simplicity of this possibility lies also
at the root of its unconvincing theoretical status, as there is no explanation
for the existence of such constant for the entire cosmic history. Herein we
explore general models of the vacuum energy density slowly evolving with the
Hubble function $H$ and/or its time derivative, $\rho_{\Lambda}(H,\dot{H})$.
Some of these models are actually well-motivated from the theoretical point of
view and may provide a rich phenomenology that could be explored in future
observations, whereas some others have more limitations. In this work, we put
them to the test and elucidate which ones are still compatible with the present
observations and which ones are already ruled out. We consider their
implications on structure formation, in combination with data on type Ia
supernovae, the Cosmic Microwave Background, the Baryonic Acoustic
Oscillations, and the predicted redshift distribution of cluster-size collapsed
structures. The relation of these vacuum models on possible evidence of
dynamical dark energy recently pointed out in the literature is also briefly
addressed.
