  A self-avoiding walk (SAW) on the square lattice is prudent if it never takes
a step towards a vertex it has already visited. Prudent walks differ from most
classes of SAW that have been counted so far in that they can wind around their
starting point. Their enumeration was first addressed by Pr\'ea in 1997. He
defined 4 classes of prudent walks, of increasing generality, and wrote a
system of recurrence relations for each of them . However, these relations
involve more and more parameters as the generality of the class increases. The
first class actually consists of partially directed walks, and its generating
function is well-known to be rational. The second class was proved to have an
algebraic (quadratic) generating function by Duchi (2005). Here, we solve
exactly the third class, which turns out to be much more complex: its
generating function is not algebraic, nor even D-finite. The fourth class --
general prudent walks -- is the only isotropic one, and still defeats us.
However, we design an isotropic family of prudent walks on the triangular
lattice, which we count exactly. Again, the generating function is proved to be
non-D-finite. We also study the asymptotic properties of these classes of
walks, with the (somewhat disappointing) conclusion that their endpoint moves
away from the origin at a positive speed. This is confirmed visually by the
random generation procedures we have designed.
