1 Introduction: topological and algebro-geometric motivations  
2 A user’s guide to $\mathbb{A}^{1}$-homotopy theory  
3 Concrete $\mathbb{A}^{1}$-weak equivalences  
4 Further computations in $\mathbb{A}^{1}$-homotopy theory  
5 Cancellation questions and $\mathbb{A}^{1}$-contractibility