  We describe the double Yangian of the general linear Lie algebra
$\mathfrak{gl}_N$ by following a general scheme of Drinfeld. This description
is based on the construction of the universal $R$-matrix for the Yangian. To
make the exposition self contained, we include the proofs of all necessary
facts about the Yangian itself. In particular, we describe the centre of the
Yangian by using its Hopf algebra structure, and provide a proof of the
analogue of the Poincar\'e-Birkhoff-Witt theorem for the Yangian based on its
representation theory. This proof extends to the double Yangian, thus giving a
description of its underlying vector space.
