In this series of lectures we will study structures on the space of knots and on the dual space, the space of knot invariants. In the first few lectures we will study a few natural operations on knots (namely, connected sum and cabling) and a natural class of invariants, the Vassiliev or finite-type invariants. This will lead to a proof that some simple topological identities (e.g., the topological equivalent of "1+1=2") imply non-trivial theorems: Wheeling (a strengthening of the existence of the Duflo isomorphism) and Wheels (a computation of all Vassiliev invariants of the unknot).

However, we will see that the result of a non-trivial operation on knots (or links) per se is always in a very restricted class of knots. To progress further, we are naturally led to the study of knotted trivalent graphs (KTGs). We will see that the Kontsevich integral is the essentially unique universal finite-type invariant of knots with a well-behaved extension to KTGs. The study of KTGs naturally leads to Turaev's shadow world, a method for describing knots or 3-manifolds using surfaces.