  In this paper we construct a Stein neighborhood basis for any compact
subvariety $A$ with strongly pseudoconvex boundary $bA$ and Stein interior
$A\backslash bA$ in a complex space $X$. This is an extension of a well known
theorem of Siu. When $A$ is a complex curve, our result coincides with the
result proved by Drinovec-Drnov\v{s}ek and Forstneri\v{c}. We shall adapt their
proof to the higher dimensional case, using also some ideas of Demailly's proof
of Siu's theorem. For embedded strongly pseudoconvex domain in a complex
manifold we also find a basis of tubular Stein neighborhoods. These results are
applied to the approximation problem for holomorphic mappings.
