You need to know: Notation 
for the number of elements in finite set S, set 
of complex numbers, real part 
and imaginary part 
of a complex number z, compact subsets of , compactly supported function 
, matrix with complex entries, eigenvalue of a square matrix, basic probability theory, independent identically distributed (i.i.d.) complex random variables, mean, variance, probability measure 
on , integration 
, uniform distribution on the unit disk in .
Background: Given an 
matrix A, let 
be the empirical spectral distribution (ESD) of its eigenvalues 
, 
. We interpret 
as a discrete probability measure on . Let 
be a sequence of random matrices. For a probability measure 
on , and continuous compactly supported function , let 
. We say that the ESD of 
converges in probability to if 
for every f and for every 
. Also, we say that ESD of converges to almost surely, if, with probability 
, 
converges to zero for all f.
The Theorem: On 30th July 2008, Terence Tao, Van Vu submitted to arxiv a paper in which they proved the following result. Let be the random matrix whose entries are i.i.d. complex random variables with mean zero and variance one. Then, the ESD of converges (both in probability and in the almost sure sense) to the uniform distribution on the unit disk.
Short context: The statement of the Theorem was known as the circular law conjecture before 2010, and now is called the circular law. It was first established by Ginibre in 1965 for random matrices with Gaussian distribution of entries. Then many authors proved it for more and more general distributions, culminating in the Theorem that establishes it in the general case.
Links: Free arxiv version of the original paper is here, journal version is here.
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