ABSTRACT: We will consider cod-1 foliations and non singular flows in 3-manifolds.  For instance if the manifold $M$ is closed, irreducible and has non trivial homology, then Gabai proved there are many finite depth foliations which reflect the topology very well - they are associated to a hierarchy of the manifold. How do these foliations relate to the geometry? First one looks for ``tight'' transverse flows. If $M$ is atoroidal then Gabai-Mosher proved
there is a flow (almost) transversal to the foliation $F$ and which is  pseudo-Anosov. Roughly this means it is transversely hyperbolic. Under these conditions $M$ is also hyperbolic and the geometry of the universal cover
$\tilde{M}$ is of major interest. We proved (with Lee Mosher) that the transverse flow is quasigeodesic - meaning it efficiently measures distances in homotopy classes - that is, they have good geometric properties. Using this we can understand asymptotic properties of leaves of the foliation $F$. The pseudo-Anosov flow generates singular stable and unstable foliations and in various cases they have excellent geometric properties (they measure distance in $\tilde{M}$ efficiently). Again using this we show that when lifted to $\tilde{M}$, the leaves of the foliation $\tilde{F}$ extend continuously to the ideal  boundary of $\tilde{M}$, which is a sphere at infinity. This  is the best geometric behavior one could expect.

If time permits we will also discuss $R$-covered Anosov foliations and when
they admit transverse pseudo-Anosov flows.