Surface wave studies in the 1960s provided the first indication that the upper mantle was radially anisotropic. Resolving the anisotropic structure is important because it may yield information on deformation and flow patterns in the upper mantle. The existing radially anisotropic models are in poor agreement. Rayleigh waves have been studied extensively and recent models show general agreement. Less work has focused on Love waves and the models that do exist are less well-constrained than are Rayleigh wave models, suggesting it is the Love wave models that are responsible for the poor agreement in the radially anisotropic structure of the upper mantle. We have adapted the waveform inversion procedure of Debayle & Ricard to extract propagation information for the fundamental mode and up to the fifth overtone from Love waveforms in the 50–250 s period range. We have tomographically inverted these results for a mantle horizontal shear wave-speed model (βh(z)) to transition zone depths. We include azimuthal anisotropy (2θ and 4θ terms) in the tomography, but in this paper we discuss only the isotropic βh(z) structure. The data set is significantly larger, almost 500 000 Love waveforms, than previously published Love wave data sets and provides ∼17 000 000 constraints on the upper-mantle βh(z) structure. Sensitivity and resolution tests show that the horizontal resolution of the model is on the order of 800–1000 km to transition zone depths. The high wave-speed roots beneath the oldest parts of the continents appear to extend deeper for βh(z) than for βv(z) as in previous βh(z) models, but the resolution tests indicate that at least parts of these features could be artefacts. The low wave speeds beneath the mid-ocean ridges fade by ∼150 km depth except for the upper mantle beneath the East Pacific Rise which remains slow to ∼250 km depth. The resolution tests suggest that the low wave speeds at deeper depths beneath the East Pacific Rise are not solely due to vertical smearing of shallow, low wave speeds. Four prominent, low wave-speed features occur at transition zone depths—one aligned along the East African Rift, one centred south of the Indian peninsula, one located south of New Zealand and one in the south Pacific Ocean coinciding with the location of the South Pacific Superswell. The low wave-speed features south of New Zealand and south of the Indian peninsula correspond spatially with the two largest negative geoid lows on Earth.