  We consider a tower of function fields F=(F_n)_{n\geq 0} over a finite field
F_q and a finite extension E/F_0 such that the sequence
\mathcal{E):=(EF_n)_{n\goq 0} is a tower over the field F_q. Then we deal with
the following: What can we say about the invariants of \mathcal{E}; i.e., the
asymptotic number of places of degree r for any r\geq 1 in \mathcal{E}, if
those of F are known? We give a method based on explicit extensions for
constructing towers of function fields over F_q with finitely many prescribed
invariants being positive, and towers of function fields over F_q, for q a
square, with at least one positive invariant and certain prescribed invariants
being zero. We show the existence of recursive towers attaining the
Drinfeld-Vladut bound of order r, for any r\geq 1 with q^r a square. Moreover,
we give some examples of recursive towers with all but one invariants equal to
zero.
