  We derive necessary and sufficient conditions for a Hill operator (i.e., a
one-dimensional periodic Schr\"odinger operator) $H=-d^2/dx^2+V$ to be a
spectral operator of scalar type. The conditions show the remarkable fact that
the property of a Hill operator being a spectral operator is independent of
smoothness (or even analyticity) properties of the potential $V$. In the course
of our analysis we also establish a functional model for periodic Schr\"odinger
operators that are spectral operators of scalar type and develop the
corresponding eigenfunction expansion.
  The problem of deciding which Hill operators are spectral operators of scalar
type appears to have been open for about 40 years.
