  Planar holomorphic systems $\dot{x}=u(x,y)$, $\dot{y}=v(x,y)$ are those that
$u=\operatorname{Re}(f)$ and $v=\operatorname{Im}(f)$ for some holomorphic
function $f(z)$. They have important dynamical properties, highlighting, for
example, the fact that they do not have limit cycles and that center-focus
problem is trivial. In particular, the hypothesis that a polynomial system is
holomorphic reduces the number of parameters of the system. Although a
polynomial system of degree $n$ depends on $n^2 +3n+2$ parameters, a polynomial
holomorphic depends only on $2n + 2$ parameters. In this work, in addition to
making a general overview of the theory of holomorphic systems, we classify all
the possible global phase portraits, on the Poincar\'{e} disk, of systems
$\dot{z}=f(z)$ and $\dot{z}=1/f(z)$, where $f(z)$ is a polynomial of degree
$2$, $3$ and $4$ in the variable $z\in \mathbb{C}$. We also classify all the
possible global phase portraits of Moebius systems $\dot{z}=\frac{Az+B}{Cz+D}$,
where $A,B,C,D\in\mathbb{C}, AD-BC\neq0$. Finally, we obtain explicit
expressions of first integrals of holomorphic systems and of conjugated
holomorphic systems, which have important applications in the study of fluid
dynamics.
