  We study the parameter space structure of degree $d \ge 3$ one complex
variable polynomials as dynamical systems acting on $\C$. We introduce and
study {\it straightening maps}. These maps are a natural higher degree
generalization of the ones introduced by Douady and Hubbard to prove the
existence of small copies of the Mandelbrot set inside itself. We establish
that straightening maps are always injective and that their image contains all
the corresponding hyperbolic systems. Also, we characterize straightening maps
with compact domain. Moreover, we give two classes of bijective straightening
maps. The first produces an infinite collection of embedded copies of the
$(d-1)$-fold product of the Mandelbrot set in the connectedness locus of degree
$d \ge 3$. The second produces an infinite collection of full families of
quadratic connected filled Julia sets in the cubic connectedness locus, such
that each filled Julia set is quasiconformally embedded.
