  Let $K$ be the fraction field of a complete discrete valuation ring, with
algebraically closed residue field of characteristic $p > 0$. This paper
studies the index of a smooth, proper $K$-variety $X$ with logarithmic good
reduction. We prove that it is prime to $p$ in `most' cases, for example if the
Euler number of $X$ does not vanish, but (perhaps surprisingly) not always. We
also fully characterise curves of genus $1$ with logarithmic good reduction,
thereby completing classical results of T. Saito and Stix valid for curves of
genus at least $2$.
