In this paper we present some of my favorite problems, all the time open, in the fixed point theory. These problems are in connection with the following two:
\begin{itemize}
\item Which properties have the fixed point equations for which an iterative algorithm is convergent?
\item Let us have a fixed point theorem, $T$, and an operator $f$ (single or multivalued) which does not satisfy the conditions in $T$. In which conditions the operator $f$ has an invariant subset $Y$ such that the restriction of $f$ to $Y$, $f|_{Y}$, satisfies the conditions of $T$?
\end{itemize}