The growth of buoyant plumes in the presence of stratification (N) and rotation (f) is studied and illustrated with a number of numerical experiments of convection induced by a localized source of buoyancy at the lower boundary of a linearly stratified fluid. The presence of stratification constrains the convection in the vertical giving rise to an equilibrium-spreading layer which receives the rising mass of plume fluid; the plume can be divided into an upper, mass-source driven anticyclone and a lower, buoyancy-source (F) driven cyclone. With N/f large, the plume's rise-height is set by the classical non-rotating scaling lN = (F/N3)1/4. Physically motivated scaling laws invoke angular momentum constraints and indicate the fundamental role played by rotation, which sets the scale lf = (F/f3)1/4. The lateral spread of the upper-level anticyclone is constrained by rotation: for times greater than f−1 the anticyclone grows laterally at a rate which is essentially independent of N, and given by lf(ft)1/3; the ratio of the lateral scale of the anticyclone to its vertical scale (aspect ratio) is proportional to N/f. The cyclone's lateral scale is lf, and the strong cyclonic flow scales like flf. An enhanced lateral mixing is suggested to occur in the cyclone along slanted angular momentum and isopycnal surfaces, which become closely aligned. On a much longer time scale, the scaling suggests that the lateral growth of the upper level anticyclone is arrested by its interaction with the lower level cyclone; a baroclinic instability is expected to detach the anticyclone from the source after a time of order ft ≈ 100N/f.