Virtual knot theory studies knots and links embedded in thickened surfaces. This is a fundamental case of knots and links in three dimensional manifolds, and it includes embeddings in the three dimensional sphere, since knots and links in a thickened two dimensional sphere are the same topologically as their embeddings in the three sphere. We explain a diagrammatic and combinatorial approach to these problems. By using diagrams in the plane, or on the two sphere using virtual crossings we can represent all virtual knots up to 1-handle stabilization in their thickened surfaces. The diagrammatic theory leads to the construction of many new invariants and to the reconsideration of known invariants. The talk will introduce these structures with many examples, and it will be self-contained.