  We study spin transport in the one-dimensional anisotropic S = 1 Heisenberg
model. Particular emphasis is given to dynamics at infinite temperature, where
current autocorrelations and spatio-temporal correlation functions are obtained
by means of an efficient pure-state approach based on the concept of
typicality. Our comprehensive numerical analysis unveils that high-temperature
spin transport is diffusive in the easy-axis regime for strong exchange
anisotropies. This finding is based on the combination of numerous signatures,
such as (i) Gaussian spreading of correlations, (ii) a time-independent
diffusion coefficient, (iii) power-law decay of equal-site correlations, (iv)
exponentially decaying long-wavelength modes, and (v) Lorentzian line shapes of
the dynamical structure factor. Moreover, we provide evidence that some of
these signatures are not exclusively restricted to the infinite-temperature
limit, but can persist at lower temperatures as well, where we complement our
results by additional quantum Monte Carlo simulations of large systems. In
contrast to the easy-axis regime, we show that in the case of an isotropic
chain, the signatures (i) - (v) are much less pronounced or even entirely
absent, suggesting the existence of anomalous spin transport despite the
nonintegrability of the model. Eventually, upon introducing a random on-site
magnetic field, we observe a breakdown of diffusion and distinctly slower
dynamics. In particular, our results exhibit qualitative similarities to
disordered spin-1/2 chains and might be consistent with the onset of many-body
localization in the S = 1 model for sufficiently strong disorder.
