This paper introduces the theory of virtual knots, a generalization of classical knot theory discovered by the author in 1996. It covers basic definitions, fundamental properties, and examples. The paper is organized into several sections: 1. **Introduction**: Provides an overview of virtual knot theory, including the definition of virtual knots and their diagrams, and the motivation from knots in thickened surfaces and Gauss codes. 2. **Virtual Knot Diagrams and Moves**: Defines virtual knots using diagrams and moves, including classical Reidemeister moves, shadowed Reidemeister moves, and a triangle move that relates two virtual crossings and one classical crossing. 3. **Basic Results**: Discusses the reconstruction properties of Gauss codes and how they can be used to identify virtual knots. 4. **Fundamental Group and Quandle**: Introduces the fundamental group and quandle (or rack) of virtual knots, showing that some virtual knots have trivial fundamental groups. 5. **Non-trivial Virtual Knots**: Examples of non-trivial virtual knots with trivial fundamental groups and trivial Jones polynomials are provided. 6. **Quantum Link Invariants**: Extends quantum link invariants to virtual knots and links, including the bracket polynomial and the Jones polynomial. 7. **Vassiliev Invariants**: Discusses Vassiliev invariants for virtual knots and links, defining graphical finite type and proving that weight systems are finite for virtual Vassiliev invariants. The paper also includes acknowledgments and detailed proofs of various theorems and lemmas, providing a comprehensive introduction to virtual knot theory.This paper introduces the theory of virtual knots, a generalization of classical knot theory discovered by the author in 1996. It covers basic definitions, fundamental properties, and examples. The paper is organized into several sections: 1. **Introduction**: Provides an overview of virtual knot theory, including the definition of virtual knots and their diagrams, and the motivation from knots in thickened surfaces and Gauss codes. 2. **Virtual Knot Diagrams and Moves**: Defines virtual knots using diagrams and moves, including classical Reidemeister moves, shadowed Reidemeister moves, and a triangle move that relates two virtual crossings and one classical crossing. 3. **Basic Results**: Discusses the reconstruction properties of Gauss codes and how they can be used to identify virtual knots. 4. **Fundamental Group and Quandle**: Introduces the fundamental group and quandle (or rack) of virtual knots, showing that some virtual knots have trivial fundamental groups. 5. **Non-trivial Virtual Knots**: Examples of non-trivial virtual knots with trivial fundamental groups and trivial Jones polynomials are provided. 6. **Quantum Link Invariants**: Extends quantum link invariants to virtual knots and links, including the bracket polynomial and the Jones polynomial. 7. **Vassiliev Invariants**: Discusses Vassiliev invariants for virtual knots and links, defining graphical finite type and proving that weight systems are finite for virtual Vassiliev invariants. The paper also includes acknowledgments and detailed proofs of various theorems and lemmas, providing a comprehensive introduction to virtual knot theory.