Despite the fact that a rigid Λ-term is a fundamental building block of the concordance ΛCDM model, we show that a large class of cosmological scenarios with dynamical vacuum energy density ρΛ and/or gravitational coupling G, together with a possible non-conservation of matter, are capable of seriously challenging the traditional phenomenological success of the ΛCDM. In this paper, we discuss these 'running vacuum models' (RVM's), in which ρΛ=ρΛ(H) consists of a nonvanishing constant term and a series of powers of the Hubble rate. Such generic structure is potentially linked to the quantum field theoretical description of the expanding Universe. By performing an overall fit to the cosmological observables SNIa+BAO+H(z)+LSS+BBN+CMB (in which the WMAP9, Planck 2013 and Planck 2015 data are taken into account), we find that the class of RVM's appears significantly more favored than the ΛCDM, namely at an unprecedented level of ≳4.2σ. Furthermore, the Akaike and Bayesian information criteria confirm that the dynamical RVM's are strongly preferred as compared to the conventional rigid Λ-picture of the cosmic evolution.