Starting with the problem of the title -- in algebraic geometry, this is known as the Zariski Cancellation Problem -- we will show how it touches upon a number of deep problems in mathematics. Classification of contractible manifolds (and an historic mis-proof of the Poincare conjecture), non-reductive group actions and quotients, classical invariant theory and Hilbert's 14th problem, the automorphism group of affine space, reinterpretations of singularities, moduli of algebraic vector bundles, and probing the extent to which detailed algebraic geometry can be captured by new techniques from algebraic topology ... all of these emerge naturally, and at heart are rather simple to explain, at least conceptually. Indeed, our basic object of study is quite friendly: free additive group actions on affine space. Pretty examples abound. This draws on joint work with Aravind Asok and with Frances Kirwan.