  We study holomorphic Legendrian curves in the standard complex contact
structure on the projectivised cotangent bundle $X=\mathbb P(T^*Z)$ of a
complex manifold $Z$ of dimension at least $2$. We provide a detailed analysis
of Legendrian curves degenerating to vertical curves and obtain several
approximation and general position theorems. In particular, we prove that any
vertical holomorphic curve $M\to X$ from a compact bordered Riemann surface $M$
can be deformed to a horizontal Legendrian curve by an arbitrarily small
deformation. A similar result is proved in the parametric setting, provided
that all vertical curves under consideration are nondegenerate. Stronger
results are obtained when the base $Z$ is an Oka manifold or a Stein manifold
with the density property. Finally, we establish basic and 1-parametric
h-principles for holomorphic Legendrian curves in $X$.
