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We describe a construction of the cyclotomic structure on topologicalHochschild homology ($THH$) of a ring spectrum using the Hill-Hopkins-Ravenelmultiplicative norm. Our analysis takes place entirely in the category ofequivariant orthogonal spectra, avoiding use of the B\"okstedt coherencemachinery. We are able to define versions of topological cyclic homology ($TC$)and TR-theory relative to a cyclotomic commutative ring spectrum $A$. Wedescribe spectral sequences computing this relative theory $_ATR$ in terms of$TR$ over the sphere spectrum and vice versa. Furthermore, our constructionpermits a straightforward definition of the Adams operations on $TR$ and $TC$.

Topological cyclic homology via the norm