Finite subgroups of $\operatorname{Aut}(\mathbb{P}^1)$ I would like to know all finite subgroups of $\operatorname{Aut}(\mathbb{P}^1)$. 
I am aware that any automorphism of $\mathbb{P}^1$ is given by a Möbius transformation $$
z\mapsto\frac{az+b}{cz+d}
$$
and thus there is an identification 
$$
\operatorname{Aut}(\mathbb{P}^1)\cong \operatorname{PSL}(2,\mathbb{C})\cong \operatorname{SO}(3, \mathbb{C}).
$$ 
I thought this solved the question, but what I know is the classification of finite subgroups of real orthogonal group $\operatorname{SO}(3, \mathbb{R})$.
 A: So pointed out by Simon Henry, $\operatorname{PSL}(2,\mathbb{C})\cong \operatorname{SO}(3, \mathbb{R})$, not $\operatorname{SO}(3, \mathbb{C})$. Recall that $SO(3, \mathbb{R})$ is the set of orientation-preserving isometries of $\mathbb{R}^3$, and hence $S^2$ (which is homeomorphic to $\mathbb{P}^1$).
The finite subgroups of $SO(3, \mathbb{R})$ have been classified, see the groupprops wikipage for example. I will briefly outline them below.
Note that $SO(2, \mathbb{R})$ naturally includes in $SO(3, \mathbb{R})$. More  precisely, the map 
\begin{align*}
SO(2, \mathbb{R}) &\to SO(3, \mathbb{R})\\
A &\mapsto \begin{bmatrix} A & 0\\ 0 & 1\end{bmatrix}
\end{align*} 
gives an isomorphism between $SO(2, \mathbb{R})$ and a subgroup of $SO(3, \mathbb{R})$. Therefore, any subgroup of $SO(2, \mathbb{R})$ can be realised as a subgroup of $SO(3, \mathbb{R})$; in particular, every finite subgroup of $SO(2, \mathbb{R})$ will give a finite subgroup $SO(3, \mathbb{R})$. Note that the elements of $SO(2, \mathbb{R})$ are simply the rotation matrices and so we have $SO(2, \mathbb{R}) \cong S^1$. The only finite subgroups of $S^1$ are finite cyclic subgroups, so $SO(3, \mathbb{R})$ has such subgroups.
Another infinite family of finite subgroups of $SO(3, \mathbb{R})$ are the dihedral groups $D_{2n}$, which are the symmetries of a regular $n$-gon in a plane. Although these symmetries leave the plane fixed, they do not come from elements of $SO(2, \mathbb{R})$ as above because reflections swap the sides of the plane.
The remaining finite subgroups of $SO(3, \mathbb{R})$ rely on the classification of platonic solids. There is a subgroup isomorphic to $A_4$ given by the symmetries of a tetrahedron, a subgroup isomorphic to $S_4$ given by the symmetries of a cube or octahedron, and a subgroup isomorphic to $A_5$ given by the symmetries of an icosahedron or dodecahedron. Note, the platonic solids which appear in pairs are dual polyhedra (a tetrahedron is self-dual).
In addition to all of the finite subgroups discussed above, we must also consider their subgroups. In doing so, we see that all of them have already been mentioned${}^1$. Therefore, a finite subgroup of $SO(3, \mathbb{R})$ is isomorphic to one of the following: $\mathbb{Z}/n\mathbb{Z}$, $D_{2n}$, $S_3$, $A_4$, $S_4$, $A_5$.

${}^1$ I initially thought $S_3$ (a subgroup of both $S_4$ and $A_5$) was missing, but as pointed out by pjs36, $S_3 \cong D_6$.
A: (I am not allowed to add any comments, so I add an answer.) In contrast to what has been said here, the groups $\mathrm{PSL}(2,\mathbf C)$ and $\mathrm{SO}(3,\mathbf R)$ are not naturally isomorphic. The topological group $\mathrm{PSL}(2,\mathbf C)$ is not compact. Indeed, the sequence of matrices
$$
\left\{
\begin{pmatrix}
1&n\\0&1
\end{pmatrix} 
\right\}_{n\in\mathbf N}$$
in $\mathrm{PSL}(2,\mathbf C)$ does not have a convergent subsequence. The group $\mathrm{SO}(3,\mathbf R)$, however, is compact, being a topological subspace of the cube of the unit sphere in $\mathbf R^3$. Therefore, the groups $\mathrm{PSL}(2,\mathbf C)$ and $\mathrm{SO}(3,\mathbf R)$ are not isomorphic as topological groups, and definitely not naturally isomorphic. A more involved argument probably shows that they are not isomorphic as abstract groups, but that is not to the point here.
What is relevant, is that $\mathrm{PSL}(2,\mathbf C)$ contains the compact subgroup $\mathrm{PSU}(2,\mathbf C)$. Moreover, any finite, or compact for that matter, subgroup of $\mathrm{PSL}(2,\mathbf C)$ is conjugate to a subgroup of $\mathrm{PSU}(2,\mathbf C)$ (cf. this question). Now, the latter group is isomorphic to $\mathrm{SO}(3,\mathbf R)$ (cf. wikipedia), and the answer of Michael Albanese applies.
A: There's an easy proof of the classification of finite subgroups  of ${\rm PGL}(2, \mathbb{C})$ in Dolgachev's notes: McKay Correspondence
You can also search for Blichfeldt's book: Finite Collineation Groups.
