Major arcs for Goldbach's theorem
arxiv.org
arxiv.org
Considerable work has been done on the weak Goldbach conjecture
culminating in a 2013 claimed proof by Harald Helfgott.
For the casual reader, there is a great novel by Apostolos Doxiadis based around Goldbach's conjecture: http://en.wikipedia.org/wiki/Uncle_Petros_and_Goldbach%27s_C...
(Doxiadis's "Logicomix" is also recommended)
Mathoverflow discussion: http://mathoverflow.net/questions/130545