We study the natural question of constructing pseudorandom generators (PRGs)
for low-degree polynomial threshold functions (PTFs). We give a PRG with
seed-length log n/eps^{O(d)} fooling degree d PTFs with error at most eps.
Previously, no nontrivial constructions were known even for quadratic threshold
functions and constant error eps. For the class of degree 1 threshold functions
or halfspaces, we construct PRGs with much better dependence on the error
parameter eps and obtain the following results.

1) A PRG with seed length O(log n log(1/eps)) for eps > 1/poly(n).

2) A PRG with seed length O(log n) for eps > 1/poly(log n).

Previously, only PRGs with seed length O(log n log^2(1/eps)/eps^2) were known
for halfspaces. We also obtain PRGs with similar seed lengths for fooling
halfspaces over the n-dimensional unit sphere.

The main theme of our constructions and analysis is the use of invariance
principles to construct pseudorandom generators. We also introduce the notion
of monotone read-once branching programs, which is key to improving the
dependence on the error rate eps for halfspaces. These techniques may be of
independent interest.