A long-standing question in the field of superconductivity is whether pairing of electrons can arise in some cases as a result of magnetic interactions instead of electron-phonon-induced interactions as in the conventional Bardeen-Cooper-Schrieffer theory 1 . A major challenge to the idea of magnetically mediated superconductivity has been the dramatically different behaviour of the cerium and ytterbium heavy-fermion compounds. The cerium-based systems are often found to be superconducting 1-6 , in keeping with a magnetic pairing scenario, but corresponding ytterbium systems, or hole analogues of the cerium systems, are not. Despite searches over two decades there has been no evidence of heavy-fermion superconductivity in an ytterbium system, casting doubt on our understanding of the electron-hole parallelism between the cerium and the ytterbium compounds. Here we present the first empirical evidence that superconductivity is indeed possible in an ytterbium-based heavy-fermion system. In particular, we observe a superconducting transition at T c = 80 mK in high-purity single crystals of YbAlB 4 in the new structural β phase 7 . We also observe a novel type of non-Fermi-liquid state above T c that arises without chemical doping, in zero applied magnetic field and at ambient pressure, establishing β-YbAlB 4 as a unique system showing quantum criticality without external tuning.First we present the bulk magnetic and electronic properties of β-YbAlB 4 , a new morphology of the previously known α-YbAlB 4 (refs 7,8). Shown in Fig. 1a is the orthorhombic crystal structure of β-YbAlB 4 and the temperature dependence of the d.c. magnetic susceptibility χ = M/H. Here, M and H represent the magnetization and external field, respectively. The magnetic susceptibility shows the strong uniaxial anisotropy of an Ising system with moments aligned along the c axis. Above 100 K the c-axis susceptibility has a Curie-Weiss form χ c (T ) = C/(T − θ W ), with θ W ∼ −210 K and a Curie constant C corresponding to an effective Ising moment µ eff = g J J Z ∼ 3.1 µ B , where g J is the Landé g factor and J Z is the c-axis component of the total angular momentum. The in-plane susceptibility, on the other hand, is almost temperature independent, showing a weak peak around 200 K.Shown in Fig. 1b is the temperature dependence of the in-plane resistivity, ρ ab , along with the estimated 4f -electron contribution ρ m (defined in the figure caption), which shows a coherence peak at about 250 K. The low residual resistivity ρ ab (0) ∼ 0.4 µ cm and correspondingly high residual resistivity ratio, ρ ab (300 K)/ρ ab (0) ∼ 300, suggest that the electronic mean free path is of the order of 0.1 µm.In contrast to most other heavy-fermion compounds, the resistivity does not show a Fermi liquid (FL) regime characterized by a T 2 temperature variation (Fig. 1b). As shown in Fig. 1b insets, ρ ab is linear between 4 and 1 K and varies as T 1.5 below T 0 ∼ 1 K down to 80 mK. Below 80 mK our highest-purity samples are superconducting (Fig. 1b, insets). We shall...