The celebrated T1 theorem due to David and Journ\’{e} gives a necessary and sufficient condition for $L^2$ boundedness of singular integral operators $T$. This was extended by Villarroya in 2015 to obtain the compactness of $T$. However, the $T1$ theorem to deduce compactness of multilinear singular integrals has been an open problem for ten years. In this talk we solve this long standing problem. Our main approaches consist of the following new ingredients: (i) a dyadic 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.
