We review pronounced non-Fermi-liquid (NFL) effects in the low-temperature\ normal state of the heavy-fermion superconductor UBe₁₃\ (T_c\approx 0.9 K). We argue that these NFL effects may presumably be related to short-range antiferromagnetic (AF) correlations which are manifested in the superconducting (SC) state by a ``line of thermodynamic anomalies'', B^\ast (T), between T\approx 0.7 K (B=0) and B\approx 4 T\ (T \rightarrow 0). These anomalies are shown to mark the precursor of the lower of the two phase transitions (at T_c1\ and T_c2) in U_1-xTh_xBe₁₃, x_c1\approx 0.019 < x<x_c2  \approx 0.0455. For x_c2 < x < x_c < 0.07, a single SC transition is stated which due to thermal expansion, \alpha (T), and specific heat, C(T), measurements, coincides with this lower transition at T_c2. We discuss two possible scenarios both of which imply an intimate interrelation of superconductivity with the symmetry-broken state that forms below T_c2. Finally, we address two other lines of thermal expansion anomalies in the T--x phase diagram of U_1-xTh_xBe₁₃\ which show an only weak dependence on magnetic field: (i) A positive \alpha (T) peak, along with a C(T) peak, is found in pure UBe₁₃\ at T_max\approx 2 K. Upon Th doping, T_max(x) is depressed linearly and vanishes close to the upper critical Th concentration x_c2\ at which the two phase transitions merge to one. (ii) A negative anomaly in \alpha (T) develops for x>x_c2 at T_min. T_min(x) increases by more than a factor of two when raising the Th content to x=0.1. Similar to previous results by Aliev et al. , [F.G. Aliev et al. ,  J. Phys.: Condens. Matter 8, 9807 (1995)], an almost temperature-independent non-linear susceptibility, \chi ⁽³⁾(T), is found for U_0.9Th_0.1Be₁₃, at striking variance to \chi ⁽³⁾(T) for pure UBe₁₃. The implications of this observation for the assignment of the valence state of Uranium at x=0 and x=0.1 are also addressed.

PACS numbers: 74.25.Bt, 74.70.Tx