  In this paper, we prove the existence of a countable family of regular
spherically symmetric self-similar solutions to focusing energy super-critical
semi-linear wave equations \begin{equation*}
  \partial_{tt}u-\Delta u=|u|^{p-1}u \qquad \text{in} \,\, \mathbb{R}^{N},
\end{equation*} where $N\geq 3$, $1+\frac{4}{N-2}<p$, and, if $N\geq 4$, $p
\leq 1+\frac{4}{N-3}$. This was previously known only in the case $N=3$, for
integer $p$ (see Bizo\'{n}, Maison and Wasserman \cite{BMW}). We also study the
asymptotics of these solutions.
