I will describes Hopf's classical construction of some essential maps between spheres.  These maps possess some curious equivariant properties.  I will use these examples to motivate the study of motivic homotopy theory, and to propose new homotopy theories that ought to be even more interesting than motivic homotopy theory.

I will also describe some recent stunning applications of motivic homotopy theory to classical homotopy theory.  Why should a homotopy theory of algebraic varieties be so useful for studying ordinary topological spaces?  I will make a first attempt at answering this philosophical question by giving a topological model for cellular 2-complete stable C-motivic homotopy theory.