Polaritons-hybrid light-matter excitations-enable nanoscale control of light. Particularly large polariton field confinement and long lifetimes can be found in graphene and materials consisting of two-dimensional layers bound by weak van der Waals forces(1,2) (vdW materials). These polaritons can be tuned by electric fields(3,4) or by material thickness(5), leading to applications including nanolasers(6), tunable infrared and terahertz detectors(7), and molecular sensors(8). Polaritons with anisotropic propagation along the surface of vdW materials have been predicted, caused by in-plane anisotropic structural and electronic properties(9). In such materials, elliptic and hyperbolic in-plane polariton dispersion can be expected (for example, plasmon polaritons in black phosphorus(9)), the latter leading to an enhanced density of optical states and ray-like directional propagation along the surface. However, observation of anisotropic polariton propagation in natural materials has so far remained elusive. Here we report anisotropic polariton propagation along the surface of alpha-MoO3, a natural vdW material. By infrared nano-imaging and nano-spectroscopy of semiconducting alpha-MoO3 flakes and disks, we visualize and verify phonon polaritons with elliptic and hyperbolic in-plane dispersion, and with wavelengths (up to 60 times smaller than the corresponding photon wavelengths) comparable to those of graphene plasmon polaritons and boron nitride phonon polaritons(3-5). From signal oscillations in real-space images we measure polariton amplitude lifetimes of 8 picoseconds, which is more than ten times larger than that of graphene plasmon polaritons at room temperature(10). They are also a factor of about four larger than the best values so far reported for phonon polaritons in isotopically engineered boron nitride(11) and for graphene plasmon polaritons at low temperatures(12). In-plane anisotropic and ultra-low-loss polaritons in vdW materials could enable directional and strong light-matter interactions, nanoscale directional energy transfer and integrated flat optics in applications ranging from biosensing to quantum nanophotonics.