Complex dynamics is the study of iteration of holomorphic functions on the Riemann sphere. In this thesis, we will focus on dynamical systems obtained by iterating rational functions, that is iterating a holomorphic endomorphism of the Riemann sphere $f : {\mathbb C}_\infty \to {\mathbb C}_\infty$. The family of iterates $\{f^n; n \ge 1\}$ divides ${\mathbb C}_\infty$ into two complementary sets; the Fatou set $F$ , that is the set of points in the neighbourhood of which the family $\{f^n; n \ge 1\}$ is normal, and the Julia set $J$, which is its complement. Each such map has fixed points, which are classified into five types according to their multiplier. The main goal of this thesis is to locally linearize $f$ in some neighbourhood of fixed points and describe the behaviour of orbits in their vicinity. We prove that locally near a fixed point $\zeta$, $f$ is conformally conjugate to a linear map $z \mapsto \lambda z$ in the case of attracting, repelling and some irrationally indifferent fixed points. In the case of superattracting fixed points f is conjugate to $z \mapsto z^p$, and in the case of rationally indifferent ones it is conjugate to $z \mapsto z+1$ in some disk with $\zeta$ on its boundary.
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