  We develop the properties of the $n$-th sequential topological complexity
$TC_n$, a homotopy invariant introduced by the third author as an extension of
Farber's topological model for studying the complexity of motion planning
algorithms in robotics. We exhibit close connections of $TC_n(X)$ to the
Lusternik-Schnirelmann category of cartesian powers of $X$, to the cup-length
of the diagonal embedding $X\hookrightarrow X^n$, and to the ratio between
homotopy dimension and connectivity of $X$. We fully compute the numerical
value of $TC_n$ for products of spheres, closed 1-connected symplectic
manifolds, and quaternionic projective spaces. Our study includes two
symmetrized versions of $TC_n(X)$. The first one, unlike Farber-Grant's
symmetric topological complexity, turns out to be a homotopy invariant of $X$;
the second one is closely tied to the homotopical properties of the
configuration space of cardinality-$n$ subsets of $X$. Special attention is
given to the case of spheres.
