  This paper interprets Hesselholt and Madsen's real topological Hochschild
homology functor THR in terms of the multiplicative norm construction. We show
that THR satisfies cofinality and Morita invariance, and that it is suitably
multiplicative. We then calculate its geometric fixed points and its Mackey
functor of components, and show a decomposition result for group-algebras.
Using these structural results we determine the homotopy type of
THR($\mathbb{F}_p$) and show that its bigraded homotopy groups are polynomial
on one generator over the bigraded homotopy groups of $H\mathbb{F}_p$. We then
calculate the homotopy type of THR($\mathbb{Z}$) away from the prime $2$, and
the homotopy ring of the geometric fixed-points spectrum
$\Phi^{\mathbb{Z}/2}$THR($\mathbb{Z}$).
