Litmust test for existence of a function from irregular shape to a square

Question

Let's suppose I have an simple irregular shape A in $\mathbb{R^{2}}$ and I am hoping to map this irregular shape A in $\mathbb{R}^{2}$ to a square B in $\mathbb{R}^{2}$.

Is there a litmus test or a theorem which would allow me to determine if such a function exists?