This paper introduces virtual knot theory, a generalization of classical knot theory. It defines virtual knots using diagrams with virtual crossings, which are not actual crossings but artifacts of the diagram. Virtual knots are defined by their Gauss codes and are invariant under generalized Reidemeister moves. The paper discusses the fundamental group, quandles, and other invariants of virtual knots. It shows that some virtual knots have trivial fundamental groups and Jones polynomials, and that virtual knots can be distinguished from their mirror images by these invariants. The paper also extends quantum link invariants to virtual knots, introducing virtual framing and a new invariant, $\overline{Z}(K)$, which depends on infinitely many variables. It discusses Vassiliev invariants and their properties for virtual knots, and concludes with open problems in the field. The paper is dedicated to the memory of Francois Jaeger.This paper introduces virtual knot theory, a generalization of classical knot theory. It defines virtual knots using diagrams with virtual crossings, which are not actual crossings but artifacts of the diagram. Virtual knots are defined by their Gauss codes and are invariant under generalized Reidemeister moves. The paper discusses the fundamental group, quandles, and other invariants of virtual knots. It shows that some virtual knots have trivial fundamental groups and Jones polynomials, and that virtual knots can be distinguished from their mirror images by these invariants. The paper also extends quantum link invariants to virtual knots, introducing virtual framing and a new invariant, $\overline{Z}(K)$, which depends on infinitely many variables. It discusses Vassiliev invariants and their properties for virtual knots, and concludes with open problems in the field. The paper is dedicated to the memory of Francois Jaeger.