We extend the Daugavet property (and a perfect version) to transfinite cardinals and provide a number of examples. We characterise the transfinite Daugavet $C(K)$ spaces in terms of a cardinal index $\mathfrak r(K)$, which generalises the notion of the reaping number of a Boolean algebra. We study several inheritance results of the transfinite Daugavet properties by almost isometric ideals, absolute sums, and tensor product spaces with a number of applications, including the classification of these properties for $L_1(\mu)$ and $L_\infty(\mu)$ spaces. The perfect version of the Daugavet property for $\omega$ is also characterised in the space of Lipschitz functions $\mathop{Lip}(M)$. Joint work with J. Langemets, M. Martin and A. Rueda Zoca.