Criteria for a complex space to be hyperbolic, hyperbolically
imbedded, taut or tautly imbedded are presented. In particular, 
the following generalization of theorems by Eastwood and Kobayashi 
is produced by replacing the requirement of hyperbolicity of the 
image space by normality of the mapping:
Let  $f:X\to Z  $ be a normal map between complex spaces $X$ and $Z$. 
If either (1) there is an open cover $\{V_
\alpha \}$ of $Z$ such that each connected component of
$f^{-1}(V_ \alpha)$ is hyperbolic or (2) for every $z\in Z$ each
connected component of $f^{-1}(z)$ is  compact hyperbolic, then
$X$ is hyperbolic.
 The following common generalization of results of Zaidenberg and 
Abate is also established:   A complex subspace $X$ of a complex space $Y$
is hyperbolically imbedded in $Y$ if the inclusion map from $X$ to $Y$ is
normal.