  This paper contains several results concerning circle action on
almost-complex and smooth manifolds. More precisely, we show that, for an
almost-complex manifold $M^{2mn}$(resp. a smooth manifold $N^{4mn}$), if there
exists a partition $\lambda=(\lambda_{1},...,\lambda_{u})$ of weight $m$ such
that the Chern number $(c_{\lambda_{1}}... c_{\lambda_{u}})^{n}[M]$ (resp.
Pontrjagin number $(p_{\lambda_{1}}... p_{\lambda_{u}})^{n}[N]$) is nonzero,
then \emph{any} circle action on $M^{2mn}$ (resp. $N^{4mn}$) has at least $n+1$
fixed points. When an even-dimensional smooth manifold $N^{2n}$ admits a
semi-free action with isolated fixed points, we show that $N^{2n}$ bounds,
which generalizes a well-known fact in the free case. We also provide a
topological obstruction, in terms of the first Chern class, to the existence of
semi-free circle action with \emph{nonempty} isolated fixed points on
almost-complex manifolds. The main ingredients of our proofs are Bott's residue
formula and rigidity theorem.
