  The higher topological complexity of a space $X$, $\text{TC}_r(X)$,
$r=2,3,\ldots$, and the topological complexity of a map $f$, $\text{TC}(f)$,
have been introduced by Rudyak and Pave\v{s}i\'{c}, respectively, as natural
extensions of Farber's topological complexity of a space. In this paper we
introduce a notion of higher topological complexity of a map~$f$,
$\text{TC}_{r,s}(f)$, for $1\leq s\leq r\geq2$, which simultaneously extends
Rudyak's and Pave\v{s}i\'{c}'s notions. Our unified concept is relevant in the
$r$-multitasking motion planning problem associated to a robot devise when the
forward kinematics map plays a role in $s$ prescribed stages of the motion
task. We study the homotopy invariance and the behavior of $\text{TC}_{r,s}$
under products and compositions of maps, as well as the dependence of
$\text{TC}_{r,s}$ on $r$ and $s$. We draw general estimates for
$\text{TC}_{r,s}(f\colon X\to Y)$ in terms of categorical invariants associated
to $X$, $Y$ and $f$. In particular, we describe within one the value of
$\text{TC}_{r,s}$ in the case of the non-trivial double covering over real
projective spaces, as well as for their complex counterparts.
