  Farber introduced a notion of topological complexity $\TC(X)$ that is related
to robotics. Here we introduce a series of numerical invariants $\TC_n(X),
n=1,2, ...$ such that $\TC_2(X)=\TC(X)$ and $\TC_n(X)\le \TC_{n+1}(X)$. For
these higher complexities, we define their symmetric versions that can also be
regarded as higher analogs of the symmetric topological complexity.
