  Let $\mathbb{Q}_3$ be the complex 3-quadric endowed with its standard complex
conformal structure. We study the complex conformal geometry of isotropic
curves in $\mathbb{Q}_3$. By an isotropic curve we mean a nonconstant
holomorphic map from a Riemann surface into $\mathbb{Q}_3$, null with respect
to the conformal structure of $\mathbb{Q}_3$. The relations between isotropic
curves and a number of relevant classes of surfaces in Riemannian and
Lorentzian spaceforms are discussed.
