  We perform a systematic study of Gushel-Mukai varieties---quadratic sections
of linear sections of cones over the Grassmannian Gr(2,5). This class of
varieties includes Clifford general curves of genus 6, Brill-Noether general
polarized K3 surfaces of genus 6, prime Fano threefolds of genus 6, and their
higher-dimensional analogues.
  We establish an intrinsic characterization of normal Gushel-Mukai varieties
in terms of their excess conormal sheaves, which leads to a new proof of the
classification theorem of Gushel and Mukai. We give a description of
isomorphism classes of Gushel-Mukai varieties and their automorphism groups in
terms of linear algebraic data naturally associated to these varieties.
  We carefully develop the relation between Gushel-Mukai varieties and
Eisenbud-Popescu-Walter sextics introduced earlier by Iliev-Manivel and
O'Grady. We describe explicitly all Gushel-Mukai varieties whose associated EPW
sextics are isomorphic or dual (we call them period partners or dual varieties
respectively). Finally, we show that in dimension 3 and higher, period
partners/dual varieties are always birationally isomorphic.
