A new stability approach is developed for a wide class of strongly non-parallel axisymmetric flows of a viscous incompressible fluid. This approach encompasses all conical flows, and all steady and weakly unsteady disturbances, while prior studies were limited to specific flows and particular disturbances. A specially derived form of the Navier-Stokes equations allows the exact reduction of the linear stability problem to a system of ordinary differential equations. We found that disturbances originating at the boundaries of a similarity region cause a variety of steady bifurcations. Consideration of the still fluid allows disturbances to be classified into inner, outer and global modes, depending on the boundary conditions perturbed. Then we identify and study modes which cause bifurcation as the Reynolds number increases. The study provides improved understanding of (a) azimuthal symmetry breaking, (b) genesis of swirl, (c) onset of heat convection, (d) hydromagnetic dynamo, (e) hysteretic transitions, and (f) jump flow separation. We also discover and analyse two new bifurcations: (g) fold catastrophes and (h) appearance of radial oscillations in swirl-free jets. The stability analysis reveals that bifurcations (a), (c) and (f) are caused by inner perturbations, bifurcations (b), (d), (e) and (g) by outer perturbations, and bifurcation (h) by global perturbations. We deduce amplitude equations to describe the nonlinear spatiotemporal development of disturbances near the critical Reynolds numbers for (b) and (g). Disturbances switching between the basic and secondary steady states are found to grow monotonically with time.