  In this paper we solve a long standing problem about the bilinear $T1$
theorem to characterize the (weighted) compactness of bilinear
Calder\'{o}n-Zygmund operators. Let $T$ be a bilinear operator associated with
a standard bilinear Calder\'{o}n-Zygmund kernel. We prove that $T$ can be
extended to a compact bilinear operator from $L^{p_1}(w_1^{p_1}) \times
L^{p_2}(w_2^{p_2})$ to $L^p(w^p)$ for all exponents $\frac1p = \frac{1}{p_1} +
\frac{1}{p_2}>0$ with $p_1, p_2 \in (1, \infty]$ and for all weights $(w_1,
w_2) \in A_{(p_1, p_2)}$ if and only if the following hypotheses hold: (H1) $T$
is associated with a compact bilinear Calder\'{o}n-Zygmund kernel, (H2) $T$
satisfies the weak compactness property, and (H3) $T(1,1), T^{*1}(1,1),
T^{*2}(1,1) \in \mathrm{CMO}(\mathbb{R}^n)$. This is also equivalent to the
endpoint compactness: (1) $T$ is compact from $L^1(w_1) \times L^1(w_2)$ to
$L^{\frac12, \infty}(w^{\frac12})$ for all $(w_1, w_2) \in A_{(1, 1)}$, or (2)
$T$ is compact from $L^{\infty}(w_1^{\infty}) \times L^{\infty}(w_2^{\infty})$
to $\mathrm{CMO}_{\lambda}(w^{\infty})$ for all $(w_1, w_2) \in A_{(\infty,
\infty)}$. Besides, any of these properties is equivalent to the fact that $T$
admits a compact bilinear dyadic representation.
  Our main approaches consist of the following new ingredients: (i) a resulting
representation of a compact bilinear Calder\'{o}n-Zygmund operator as an
average of some compact bilinear dyadic shifts and paraproducts; (ii)
extrapolation of endpoint compactness for bilinear operators; and (iii)
compactness criterion in weighted Lorentz spaces. Finally, to illustrate the
applicability of our result, we demonstrate the hypotheses (H1)-(H3) through
examples including bilinear continuous/dyadic paraproducts, bilinear
pseudo-differential operators, and bilinear commutators.
