This paper considers the relation between theory, computation and new experiments in noncolloidal suspensions with Newtonian and viscoelastic matrices in shearing flow. One expects from creeping flow theory that a suspension of rigid spheres in a Newtonian fluid matrix at negligible Reynolds number and large Peclet number would exhibit a constant shear viscosity. However, one usually sees shear-thinning behaviour in experiments for larger volume fractions (phi > 0.3). Experiments with a Boger fluid matrix of constant shear viscosity on the other hand, show mild shear thickening at higher shear rates. Explanations for this behaviour are unsatisfying. There are also puzzles for the normal stresses in sheared suspensions. For a suspension with a Newtonian matrix in a simple shear flow various workers have found experimentally that the bulk stress field is no longer a shear plus an isotropic pressure, and that both first and second normal stress differences (N-1 and N-2) occur and are negative. Our own experiments using an open semi-circular trough to find N-2 agree reasonably well with previous experimental results and the Brady-Morris theory of 1997. However, this theory predicts that N-1 is zero, whereas it is measured to be negative but smaller in magnitude than N-2. The viscometric functions (eta, N-1 and N-2) for non-colloidal suspensions of spheres in a Boger fluid matrix were also measured. Volume fractions (phi) from 5% up to 40% were investigated. The relative viscosity (eta(r) = eta/eta(0)) and the (positive) first normal stress difference N-1 showed increases with phi which were larger than the dilute suspension theory predictions of 1 + 2.5 phi, indicating semi-dilute suspension behaviour, even down to 5% concentration. A major concern however centres on the second normal stress difference N-2. The Boger matrix fluid showed a zero second normal stress difference, and the measurements showed that N-2 was always negative for the suspensions. This agrees with the dilute suspension theoretical prediction found using the Landau-Lifshitz averaging procedure, but not with the result from the ensemble averaging method, which predicts a positive N-2. The reason for this predictive failure is suggested to lie in the treatment of the stresses induced by the far distant spheres. It is suggested that using a Brinkman approach to model the effect of the far spheres would be useful. For larger concentrations (up to 0.4) the second normal stress difference was always negative. However, we saw a switch from a positive N-1 to a negative value when the volume fraction exceeded 0.3, similar to the results of Aral and Kalyon (1997). This can be understood as a contest between the component of (positive) N-1 due to the matrix properties and the influence of the component of (negative) N-1 due to the proximity of the spheres. A suggestion for a constitutive model based on this idea is given. (C) 2014 Elsevier B.V. All rights reserved.