  We study the existence of topologically closed complex curves normalized by
bordered Riemann surfaces in complex spaces. Our main result is that such
curves abound in any noncompact complex space admitting an exhaustion function
whose Levi form has at least two positive eigenvalues at every point outside a
compact set, and this condition is essential. The proof involves a lifting
method for the boundary of the curve and a newly developed technique of gluing
holomorphic sprays over Cartan pairs in Stein manifolds whose value lie in a
complex space, with control up to the boundary of the domains. (The latter
technique is also exploited in the subsequent papers math.CV/0607185 and
math.CV/0609706.) We also prove that any compact complex curve with C^2
boundary in a complex space admits a basis of open Stein neighborhoods. In
particular, an embedded disc of class C^2 with holomorphic interior in a
complex manifold admits a basis of open polydisc neighborhoods.
