The modern reference work on the subject seems to be [1], but it spends only a page and a half on the subject of primes in multivariate quadratic polynomials (pp. 396-397). More than half this space is devoted to Iwaniec's 1974 result. The balance mentions Sarnak's application to the Problem of Apollonius and a result of "J. Cho and H. Kim" on counting primes in $\mathbb{Q}[\sqrt{-2}].$ So nothing there.
Pleasants [2] shows that, subject to a Davenport-Lewis [3] condition on the $h^*$ (a complexity measure on the cubic form part), multivariate cubic polynomials have the expected number of primes. Unfortunately this condition requires (as a necessary but insufficient condition) that there be at least 8 variables. Further, it double-counts repeated primes.
Goldoni [4] recently wrote a thesis on this general topic. His new results (Chapter 5) on the $h$ and $h^*$ invariants make it easier to use the results of Pleasants but do not extend them to cubic polynomials with fewer than 8 variables.
Of course I would be remiss in failing to mention the groundbreaking work of Heath-Brown [5], building on Friedlander & Iwaniec [6]. These results will no doubt clear the way for broader research, but so far have not been generalized.
So in short it appears that:
* Nothing further is known about primes represented by quadratic polynomials.
* Apart from $x^3+2y^3$, almost nothing is known about which primes are represented by cubic polynomials, though some results are known for how often such polynomials take on prime values provided $h^*$ and hence the number of variables is large enough.
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On the historical side, of course Fermat is responsible for the proof of the case $x^2+y^2$. I have references that say that Weber [7] and Schering [8] handled the case of (primitive) binary quadratic forms with nonsquare discriminants, but I haven't read the papers. Motohashi [9] proved that there are $\gg n/\log^2 n$ primes of the form $x^2+y^2+1$ up to $n$, apparently (?) the first such result with a constant term. He conjectured that the true number was
$$\frac{n}{(\log n)^{3/2}}\cdot\frac32\prod_{p\equiv3(4)}\left(1-\frac{1}{p^2}\right)^{-1/2}\left(1-\frac{1}{p(p-1)}\right)$$
but as far as I know the constant still has not been proved even for this special form.
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[1] Friedlander, J. and Iwaniec, H. (2010). Opera de Cribro. AMS.
[2] Pleasants, P. (1966). <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa12/aa1212.pdf">The representation of primes by cubic polynomials</a>, _Acta Arithmetica_ **12**, pp. 23-44.
[3] Davenport, H. and Lewis, D. J. (1964). "Non-homogeneous cubic equations". _Journal of the London Mathematical Society_ **39**, pp. 657-671.
[4] Goldoni, L. (2010). <a href="http://eprints-phd.biblio.unitn.it/384/1/Thesis.pdf">Prime Numbers and Polynomials</a>. Doctoral thesis, Università degli Studi di Trento.
[5] Heath-Brown, D. R. (2001). <a href="http://eprints.maths.ox.ac.uk/168/">Primes represented by $x^3 + 2y^3$</a>. _Acta Mathematica_ **186**, pp. 1-84.
[6] Friedlander, J. and Iwaniec, H. (1997). <a href="http://www.pnas.org/content/94/4/1054.abstract">Using a parity-sensitive sieve to count prime values of a polynomial</a>. _Proceedings of the National Academy of Sciences_ **94**, pp. 1054-1058.
[7] Weber, H. (1882). "Beweis des Satzes, dass, usw". _Mathematische Annalen_ **20**, pp. 301-329.
[8] Schering, E. (1909). "Beweis des Dirichletschen Satzes". Gesammelte mathematische Werke, Bd. 2, pp. 357-365.
[9] Motohashi, Y. (1969). <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa16/aa1642.pdf">On the distribution of prime numbers which are of the form $x^2+y^2+1$</a>. _Acta Arithmetica_ **16**, pp. 351-364.