  We consider a class of quantum lattice models in $1+1$ dimensions represented
as local quantum circuits that enjoy a particular "dual-unitarity" property. In
essence, this property ensures that both the evolution "in time" and that "in
space" are given in terms of unitary transfer matrices. We show that for this
class of circuits, generically non-integrable, one can compute explicitly all
dynamical correlations of local observables. Our result is exact,
non-pertubative, and holds for any dimension $d$ of the local Hilbert space. In
the minimal case of qubits ($d = 2$) we also present a classification of all
dual-unitary circuits which allows us to single out a number of distinct
classes for the behaviour of the dynamical correlations. We find
"non-interacting" classes, where all correlations are preserved, the ergodic
and mixing one, where all correlations decay, and, interestingly, also classes
that are are both interacting and non-ergodic.
