  We define and develop a homotopy invariant notion for the topological
complexity of a map $f:X \to Y$, denoted TC($f$), that interacts with TC($X$)
and TC($Y$) in the same way cat($f$) interacts with cat($X$) and cat($Y$).
Furthermore, TC($f$) and cat($f$) satisfy the same inequalities as TC($X$) and
cat($X$). We compare it to other invariants defined in the papers
[15,16,17,18,20]. We apply TC($f$) to studying group homomorphisms $f:H\to G$.
