  In this paper, we study the properties of the Eberlein convolution of
measures and introduce a twisted version of it. For functions we show that the
twisted Eberlein convolution can be seen as a translation invariant
function-valued inner product. We study its regularity properties and show its
existence on suitable sets of functions. For translation bounded measures we
show that the (twisted) Eberlein convolution always exists along subsequences
of the given sequence, and is a weakly almost periodic and Fourier
transformable measure. We prove that if one of the two measures is mean almost
periodic, then the (twisted) Eberlein convolution is strongly almost periodic.
Moreover, if one of the measures is norm almost periodic, so is the (twisted)
Eberlein convolution.
