  We construct a family of oriented extended topological field theories using
the AKSZ construction in derived algebraic geometry, which can be viewed as an
algebraic and topological version of the classical AKSZ field theories that
occur in physics. These have as their targets higher categories of symplectic
derived stacks, with higher morphisms given by iterated Lagrangian
correspondences. We define these, as well as analogous higher categories of
oriented derived stacks and iterated oriented cospans, and prove that all
objects are fully dualizable. Then we set up a functorial version of the AKSZ
construction, first implemented in this context by
Pantev-To\"en-Vaqui\'e-Vezzosi, and show that it induces a family of symmetric
monoidal functors from oriented stacks to symplectic stacks. Finally, we
construct forgetful functors from the unoriented bordism $(\infty,n)$-category
to cospans of spaces, and from the oriented bordism $(\infty,n)$-category to
cospans of spaces equipped with an orientation; the latter combines with the
AKSZ functors by viewing spaces as constant stacks, giving the desired field
theories.
