  We prove an asymptotic formula with four main terms for the fourth moment of
quadratic Dirichlet $L$-functions unconditionally. Our proof is based on the
work of Li , Soundararajan, and Soundararajan-Young. Our proof requires several
new ingredients. These include a modified large sieve estimate for quadratic
characters where we consider a fourth moment, rather than a second, as well as
observing cross cancellations between diagonal and off-diagonal terms, which
involves somewhat delicate combinatorial arguments.
