Let X be a closed subspace of a Banach space Y and J be the inclusion map. We say that the pair (X,Y) has the Daugavet property if for every rank one bounded linear operator T from X to Y the following equality \|J+T\|=1+\|T\| holds. A new characterization of the Daugavet property in terms of weak open sets is given. It is shown that the operators not fixing copies of l_1 on a Daugavet pair satisfy the Daugavet equation. Some hereditary properties are found: if X is a Daugavet space and Y is its subspace, then Y is also a Daugavet space provided X/Y has the Radon-Nikodym property; if Y is reflexive, then X/Y is a Daugavet space. Becides, we prove that if (X,Y) has the Daugavet property and Y \subset Z, then Z can be renormed so that (X,Z) possesses the Daugavet property and the equivalent norm coincides with the original one on Y.