Cannon and Thurston proved that for any $M$ closed hyperbolic 3–manifold which fibres over $S^1$ with hyperbolic fibre $S$ and pseudo-Anosov monodromy, the lift of the inclusion of $S$ in $M$ to universal covers extends to a continuous map of $B^2$ to $B^3$, called a Cannon-Thurston map, where $B^n =\mathbb{H}^n\cup S^{n-1}_{\infty}$ . The restriction to $S_{\infty}^1$ maps onto $S^2_{\infty}$ and gives an example of an equivariant $S^2$-filling Peano curve. In this talk, I will explain the prerequisite hyperbolic geometry and surface topology, and sketch the construction of the Cannon-Thurston map.