A numerical investigation is made of three-dimensional natural convection of a Boussinesq-fluid in a vertically-mounted cylindrical container. The boundary conditions are such that the wall temperature theta(Sigma) is inhomogeneous in the horizontal azimuthal direction but increases in the vertical direction. Interest is confined to flows with globallystable stratifications and with substantial azimuthal variations in thermal boundary conditions. Comprehensive numerical solutions to the Navier-Stokes equations are obtained. A variety of specific thermal boundary conditions are considered for detailed examination. Flow characteristics are described in broad ranges of principal nondimensional parameters, i.e., the vertical and horizontal Rayleigh numbers, the container aspect ratio and the Prandtl number. Three-dimensional flow patterns are constructed. For large Rayleigh numbers, the azimuthal inhomogeneity of boundary conditions is absorbed in the boundary layers. In the interior core, flow is determined mostly by the azimuthally-average temperature boundary condition. Exemplifications are made for two cases: (1) when theta(Sigma) is vertically uniform, and (2) when theta(Sigma) is a linear function of height. For both cases, the interior core is stably-stratified, and on the planes of constant height, horizontal motions are present. Vertical and horizontal profiles of major flow variables are plotted. The explicit effect of increasing the vertical gradient of theta(Sigma) on the global flow structure is delineated.