  In this paper, we study the well-posedness theory and the scattering
asymptotics for the energy-critical, Schr\"odinger equation with general
nonlinearity \begin{equation*}
  \left\{\begin{array}{l} i \partial_t u+\Delta u + f(u)=0,\ (x, t) \in
\mathbb{R}^N \times \mathbb{R}, \\ \left.u\right|_{t=0}=u_0 \in H
^1(\mathbb{R}^N), \end{array}\right. \end{equation*} where
$f:\mathbb{C}\rightarrow \mathbb{C}$ satisfies Sobolev critical growth
condition. Using contraction mapping method and concentration compactness
argument, we obtain the well-posedness theory in proper function spaces and
scattering asymptotics. This paper generalizes the conclusions in
\cite{KCEMF2006}(Invent. Math).
