  The aim of this manuscript is to study \emph{spear operators}: bounded linear
operators $G$ between Banach spaces $X$ and $Y$ satisfying that for every other
bounded linear operator $T:X\longrightarrow Y$ there exists a modulus-one
scalar $\omega$ such that $$ \|G + \omega\,T\|=1+ \|T\|. $$ To this end, we
introduce two related properties, one weaker called the alternative Daugavet
property (if rank-one operators $T$ satisfy the requirements), and one stronger
called lushness, and we develop a complete theory about the relations between
these three properties. To do this, the concepts of spear vector and spear set
play an important role. Further, we provide with many examples among classical
spaces, being one of them the lushness of the Fourier transform on $L_1$. We
also study the relation of these properties with the Radon-Nikod\'ym property,
with Asplund spaces, with the duality, and we provide some stability results.
Further, we present some isometric and isomorphic consequences of these
properties as, for instance, that $\ell_1$ is contained in the dual of the
domain of every real operator with infinite rank and the alternative Daugavet
property, and that these three concepts behave badly with smoothness and
rotundity. Finally, we study Lipschitz spear operators (that is, those
Lipschitz operators satisfying the Lipschitz version of the equation above) and
prove that (linear) lush operators are Lipschitz spear operators.
