Classical knot theory is concerned with the way a circle can be embedded in the 3-sphere or equivalently the way a torus can be embedded in the 3-sphere. The notion of knotted surfaces in the 3-sphere generalizes classical knot theory to include surfaces of higher genus. The theory of knotted surfaces in the 3-sphere not only is beautiful in its own right but also has interesting applications in computer science.

In this talk, I will give a brief introduction to knotted surface theory in the 3-space. I will start with some motivations behind the joint project with M. Paolini and G. Bellettini and then move on to relations between other embedding theories and knotted surface theory in the 3-space. Some known results and recent advances in this research area will also be discussed.