  The space of symplectic connections on a symplectic manifold is a symplectic
affine space. M. Cahen and S. Gutt showed that the action of the group of
Hamiltonian diffeomorphisms on this space is Hamiltonian and calculated the
moment map. This is analogous to, but distinct from, the action of Hamiltonian
diffeomorphisms on the space of compatible almost complex structures that
motivates study of extremal K\"ahler metrics. In particular, moment constant
connections are critical, where a symplectic connection is \textit{critical} if
it is critical, with respect to arbitrary variations, for the $L^{2}$-norm of
the Cahen-Gutt moment map. This occurs if and only if the Hamiltonian vector
field generated by its moment map image is an infinitesimal automorphism of the
symplectic connection. This paper develops the study of moment constant and
critical symplectic connections, following, to the extent possible, the analogy
with the similar, but different, setting of constant scalar curvature and
extremal K\"ahler metrics.
  It focuses on the special context of critical symplectic connections on
surfaces, for which general structural results are obtained, although some
results about the higher-dimensional case are included as well. For surfaces,
projectively flat and preferred symplectic connections are critical, and the
relations between these and other related conditions are examined in detail.
The relation between the Cahen-Gutt moment map and the Goldman moment map for
projective structures is explained.
