We settle a problem of Dujmovi\'c, Eppstein, Suderman, and Wood by showing
that there exists a function $f$ with the property that every planar graph $G$
with maximum degree $d$ admits a drawing with noncrossing straight-line edges,
using at most $f(d)$ different slopes. If we allow the edges to be represented
by polygonal paths with {\em one} bend, then $2d$ slopes suffice. Allowing {\em
two} bends per edge, every planar graph with maximum degree $d\ge 3$ can be
drawn using segments of at most $\lceil d/2\rceil$ different slopes. There is
only one exception: the graph formed by the edges of an octahedron is
4-regular, yet it requires 3 slopes. These bounds cannot be improved.