  We show that a separable C*-algebra is an inductive limits of projective
C*-algebras if and only if it has trivial shape, that is, if it is shape
equivalent to the zero C*-algebra. In particular, every contractible C*-algebra
is an inductive limit of projectives, and one may assume that the connecting
morphisms are surjective. Interestingly, an example of Dadarlat shows that
trivial shape does not pass to full hereditary sub-C*-algebra. It then follows
that the same fails for projectivity.
  To obtain these results, we develop criteria for inductive limit
decompositions, and we discuss the relation with different concepts of
approximation.
  As a main application of our findings we show that a C*-algebra is (weakly)
projective if and only if it is (weakly) semiprojective and has trivial shape.
It follows that a \ca{} is projective if and only if it is contractible and
semiprojective. This confirms a conjecture of Loring.
