The D-equivalence conjecture of Bondal and Orlov predicts that birational Calabi-Yau varieties have equivalent derived categories of coherent sheaves.  I will explain how to prove this conjecture for hyperkahler varieties of K3^[n] type (i.e. those that are deformation equivalent to Hilbert schemes of K3 surfaces).  This is joint work with Junliang Shen, Qizheng Yin, and Ruxuan Zhang.