  It is well known that the harmonic sum $H_n(1)=\sum_{k=1}^n\frac{1}{k}$ is
never an integer for $n>1$. In 1946, Erd\H{o}s and Niven proved that the nested
multiple harmonic sum $H_n(\{1\}^r)=\sum_{1\le k_1<\dots<k_r\le
n}\frac{1}{k_1\cdots k_r}$ can take integer values only for a finite number of
positive integers $n$. In 2012, Chen and Tang refined this result by showing
that $H_n(\{1\}^r)$ is an integer only for $(n,r)=(1,1)$ and $(n,r)=(3,2)$. In
this paper, we consider the integrality problem for arbitrary multiple harmonic
and multiple harmonic star sums and show that none of these sums is an integer
with some natural exceptions like those mentioned above.
