  We investigate which filters on $\omega$ can contain towers, that is, a
modulo finite descending sequence without any pseudointersection (in
$[\omega]^\omega$). We prove the following results:
  - Many classical examples of nice tall filters contain no towers (in ZFC).
  - It is consistent that tall analytic P-filters contain towers of arbitrary
regular height (simultaneously for many regular cardinals as well).
  - It is consistent that all towers generate non-meager filters, in particular
(consistently) Borel filters do not contain towers.
  - The statement "Every ultrafilter contains towers." is independent of ZFC.
  Furthermore, we study many possible logical (non)implications between the
existence of towers in filters, inequalities between cardinal invariants of
filters ($\mbox{add}^*(\mathcal F)$, $\mbox{cof}^*(\mathcal F)$,
$\mbox{non}^*(\mathcal F)$, and $\mbox{cov}^*(\mathcal F)$), and the existence
of Luzin type families (of size $\geq \omega_2$), that is, if $\mathcal F$ is a
filter then $X\subseteq [\omega]^\omega$ is an $\mathcal F$-Luzin family if
$\{A\in X:|A\setminus F|=\omega\}$ is countable for every $F\in \mathcal F$.
