  We present some new and recent algorithmic results concerning polynomial
system solving over various rings. In particular, we present some of the best
recent bounds on: (a) the complexity of calculating the complex dimension of an
algebraic set, (b) the height of the zero-dimensional part of an algebraic set
over C, and (c) the number of connected components of a semi-algebraic set. We
also present some results which significantly lower the complexity of deciding
the emptiness of hypersurface intersections over C and Q, given the truth of
the Generalized Riemann Hypothesis. Furthermore, we state some recent progress
on the decidability of the prefixes \exists\forall\exists and
\exists\exists\forall\exists, quantified over the positive integers. As an
application, we conclude with a result connecting Hilbert's Tenth Problem in
three variables and height bounds for integral points on algebraic curves.
  This paper is based on three lectures presented at the conference
corresponding to this proceedings volume. The titles of the lectures were
``Some Speed-Ups in Computational Algebraic Geometry,'' ``Diophantine Problems
Nearly in the Polynomial Hierarchy,'' and ``Curves, Surfaces, and the Frontier
to Undecidability.''
