We obtain global solutions for radiatively inefficiently accretion flows around black holes. Whether and where convection develops in a flow are self-consistently determined by mixinglength theory. The solutions can be divided into three types according to the strength of normal viscosity. Type I solutions correspond to large viscosity parameter α 0.1-they are purely advection-dominated with no convection; they have been extensively studied in the literature. Type II solutions are for moderate α 0.01, and have a three-zone structure. The inner zone is advection-dominated, the middle zone is convection-dominated and ranges from a few tens to a few thousands of gravitational radii, and the outer zone is convectively stable and outwardly matches a Keplerian disc. The net energy flux throughout the flow is inward, as in type I solutions. Type III solutions, which are for small α 0.001, consist of two zones as Abramowicz et al. suggested previously: an inner advection-dominated zone and an outer convection-dominated zone, separated at a radius of a few tens of gravitational radii. This type of solution has an outward net energy flux. In both type II and type III solutions, the radial density profile is between the 1/2 law of the self-similar convection-dominated accretion flow model and the 3/2 law of the self-similar advection-dominated accretion flow model, and the efficiency of energy release is found to be extremely low. Our results are in good agreement with those of recent numerical simulations.