  Explicit evaluations of the Tornheim-like double series in the form \[
\sum_{n,m=1}^\infty \frac{H_{n+m+s}}{nm\left( n+m+s \right)},\ s\in
\mathbb{N\cup } \left\{ 0 \right\} \] and their extensions are given.
Furthermore, series of the type \[ \sum_{m=1}^\infty
\frac{2H_{2m+1}-H_{m}}{2m\left( 2m+1 \right)} \] and some other Tornheim-like
multiple series are evaluated in terms of the zeta values.
