Mapping complex analysis examples

How do you show conformal mapping?
A conformal mapping of a domain Ω in the z-plane onto a domain Ω′ in the w-plane is provided by an analytic function z↦w:=f(z) defined in Ω and mapping Ω bijectively onto Ω′.
From general properties of such functions it follows that f′(z)≠0 in Ω. and map circles (:= circles or lines) onto circles to start with..
What is an example of a conformal map?
For example, stereographic projection of a sphere onto the plane augmented with a point at infinity is a conformal map.
One can also define a conformal structure on a smooth manifold, as a class of conformally equivalent Riemannian metrics..

How do you show conformal mapping?