  We study exact algorithms for Euclidean TSP in $\mathbb{R}^d$. In the early
1990s algorithms with $n^{O(\sqrt{n})}$ running time were presented for the
planar case, and some years later an algorithm with $n^{O(n^{1-1/d})}$ running
time was presented for any $d\geq 2$. Despite significant interest in
subexponential exact algorithms over the past decade, there has been no
progress on Euclidean TSP, except for a lower bound stating that the problem
admits no $2^{O(n^{1-1/d-\epsilon})}$ algorithm unless ETH fails. Up to
constant factors in the exponent, we settle the complexity of Euclidean TSP by
giving a $2^{O(n^{1-1/d})}$ algorithm and by showing that a $2^{o(n^{1-1/d})}$
algorithm does not exist unless ETH fails.
