What is the state-of-the-art for solving polynomials systems over fields that are not algebraically closed?

For the real field:

MR2830310 Sottile, Frank Real solutions to equations from geometry. University Lecture Series, 57. American Mathematical Society, Providence, RI, 2011.

MR2275625 Mikhalkin, Grigory Tropical geometry and its applications. International Congress of Mathematicians. Vol. II, 827–852, Eur. Math. Soc., Zürich, 2006.

MR1108621 Khovanskiĭ, A. G. Fewnomials. American Mathematical Society, Providence, RI, 1991.

MR1659509 Bochnak, Jacek; Coste, Michel; Roy, Marie-Françoise Real algebraic geometry. Springer-Verlag, Berlin, 1998.

For other fields:

MR2247966 Vakil, Ravi Schubert induction. Ann. of Math. (2) 164 (2006), no. 2, 489–512.

Also:

MR1925796 Sturmfels, Bernd Solving systems of polynomial equations. CBMS Regional Conference Series in Mathematics, 97. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2002.

where the real field is also discussed.

For the reals, I particularly like the book by Sturmfels mentioned by Alexandre Eremenko. For the rational numbers, you can hardly do better than Bjorn Poonen's book Rational Points on Varieties, which is available for browsing via his homepage.

For dimension $1$ specifically, Poonen also has a set of lecture notes on rational points on curves, although I always have trouble finding it. Moreover he has several expository articles (listed as such on his page) dealing with rational points on curves.

Restricted to the case of the field of rational numbers and dimension $1$ alone, this is a huge question. Restricting only to the field of rational numbers makes it even huger. Dropping any restrictions on the field entirely makes it well-nigh impossible to answer in full geberality...