  We prove limit theorems for random walks with $n$ steps in the
$d$-dimensional Euclidean space as both $n$ and $d$ tend to infinity. One of
our results states that the path of such a random walk, viewed as a compact
subset of the infinite-dimensional Hilbert space $\ell^2$, converges in
probability in the Hausdorff distance up to isometry and also in the
Gromov-Hausdorff sense to the Wiener spiral, as $d,n\to\infty$. Another group
of results describes various possible limit distributions for the squared
distance between the random walker at time $n$ and the origin.
