Best record toward Selberg's eigenvalue conjecture? What's the best record toward Selberg's eigenvalue conjecture:
a Maass form on $\Gamma_0(N)$ has eigenvalue greater than or equal to 1/4?
 A: \[\frac{1}{4} - \left(\frac{7}{64}\right)^2 = \frac{975}{4096} \approx 0.238037\ldots\]
as far as I know. Established in an appendix by Kim and Sarnak to Kim's "Functoriality for the exterior square of $GL_{4}$ and the symmetric fourth of $GL_{2}$", Journal of the American Math. Society, 16 (2003). 
The above is "over the rationals" (which was/is my reading of the question), in comments work on the problem for general number fields got mentioned by Kim and Shaidi, Cuspidality of symmetric powers with applications. Duke Math. J. 112 (2002)  and the more recent work of Blomer and Brumley, On the Ramanujan conjecture over number fields. Annals of Math. 174 (2011) which among others establishes the above for general number fields.
For generalizations in a different direction let me also mention Bourgain, Gamburd, Sarnak,  Generalization of Selberg's 3/16 Theorem and Affine Sieve, Acta Math. 207 (2011) establishing 'a generalization of Selberg's theorem for infinite index "congruence" subgroups of SL(2,Z).'
