  We consider some known and some new properties of the family of polynomials
introduced by Ted Suffridge in 1969. We begin by giving a brief overview of
their extremal properties in classic and more recent work. We also give a
compact form for Suffridge polynomials which matches a general pattern
discovered by Brandt. Our approach allows us to find the coefficients which
Brandt's result was not giving explicitly. This new presentation provides us
the tools to obtain an estimate of the rate of approximation of the generalized
Koebe functions by univalent polynomials. Furthermore, we consider the
presentation of Suffridge polynomials in Robertson's form and find the suiting
Robertson measure. This suggests a new way to approximate step functions by
continuous monotonic ones. We then study the lack of robustness of the
univalency of these polynomials and suggest a new family of polynomials for
which we conjecture the univalency of a subclass. Namely, we prove the quite
surprising fact that by extending the family by letting the discrete argument
in the polynomial coefficients become continuous one does not increase the set
of univalent polynomials. Only the initial polynomials remain univalent. In
this new one parameter family generalizing the Suffridge polynomials, it is
remarkable that the Suffridge polynomials are already extremal as they
correspond to the choice of the parameter set to 1; moreover the complex
Fej\'er polynomials correspond the choice of the parameter set to 0, and the
choice of the parameter set to -1 corresponds the polynomials $z+(z^N/N)$.
Remarkably, computer simulations seem to clearly indicate that the image of the
unit disc under these new polynomial mapping is a simply-connected region
bounded by a simple curve. This justifies the conjectural univalency of these
polynomials for the whole range of the parameters.
