  We develop a notion of forking for Galois-types in the context of Abstract
Elementary Classes (AECs). Under the hypotheses that an AEC $K$ is tame,
type-short, and failure of an order-property, we consider
  {\bf Definition.} Let $M_0 \prec N$ be models from $K$ and $A$ be a set. We
say that the Galois-type of $A$ over $M$ \emph{does not fork over $M_0$} iff
for all small $a \in A$ and all small $N^- \prec N$, we have that Galois-type
of $a$ over $N^-$ is realized in $M_0$.
  Assuming property (E) (see Definition 3.3) we show that this non-forking is a
well behaved notion of independence, in particular satisfies symmetry and
uniqueness and has a corresponding U-rank. We find conditions for a universal
local character, in particular derive superstability-like property from little
more than categoricity in a \big cardinal". Finally, we show that under large
cardinal axioms the proofs are simpler and the non-forking is more powerful. In
[BGKV] it is established that this notion of non-forking is the only
independence relation possible.
