In heavy-fermion systems with 4f or 5f atoms (such as Ce or U) the competition between the on-site moment compensation by the Kondo effect and the long-range RKKY interaction between localized magnetic moments leads to the possibility of either a non-magnetic or a magnetically ordered ground state. However, even in the case of no long-range magnetic order as exemplified by CeCu6, short-range dynamic intersite correlations are observed. Yet, the thermodynamic and transport properties of this alloy at very low temperatures T resemble those of a Fermi liquid (FL). Upon alloying with Au, long-range incommensurate antiferromagnetism is observed in CeCu6-x Au-x for x > 0.1. For x = 0.1 where T-N = 0, the specific heat C depends on T as C/T similar to -In(T/T-0), the magnetic susceptibility as chi similar to 1 - a root T, and the T-dependent part of the electrical resistivity as Delta rho similar to T. This is in marked contrast to the FL behaviour C/T similar to chi similar to constant, Delta rho similar to T-2. It is suggested that low-energy spin excitations are at the origin of these non-Fermi-liquid (NFL) anomalies which occur at a zero-temperature quantum phase transition. Large magnetic fields B restore FL behaviour. The low-T range of FL behaviour in C and rho extends towards higher T with increasing B, with the crossover temperature T-cr varying roughly linearly with. Apart from changing the Au concentration x, the magnetic-non-magnetic transition can be tuned by applying pressure p to antiferromagnetic samples with x > 0.1. For x = 0.3, T-N --> 0 at p(c) approximate to 8 kbar and NFL behaviour is observed in the specific heat for this critical pressure. For x = 0.2, p(c) approximate to 4 kbar where we likewise observe a logarithmic divergence of C/T and for p = 6.9 kbar we recover FL behaviour. Finally, we report on a remarkable 'universality' of C/T in the system CeCu(6-x)M(x) with M = Au, Pd, Pt: regardless of how T-N = 0 is reached in this system (alloying with different elements M, varying concentration, or applying pressure), the C/T versus In T curves are practically identical. Possible origins of NFL behaviour are discussed.