The paper describes a novel implementation of the piecewise linear interface-capturing volume-of-fluid method (PLIC-VOF) in axisymmetric cylindrical coordinates. The principal innovative feature involved in this work is that both the forward and inverse reconstruction problems are solved analytically, resulting in an appreciable speed-up in computing time in comparison with an iterative approach. All reconstruction formulae are introduced explicitly, and an example illustrating their derivation is included for clarity. The numerical implementation of the PLIC-VOF interface tracking method developed here is described in detail, as well as its coupling with the 3D incompressible Navier-Stokes solver PSIBOIL, which features a finite-volume approach based on a fixed, rectangular grid. This coupling includes a method to calculate the surface tension force within an axisymmetric VOF framework by means of height functions. The method is first verified to ensure its correct implementation in the code, and to evaluate its performance, and several advection tests are employed to demonstrate successful solution of the basic transport problem. It is shown that convergence to equilibrium may be achieved for static problems, involving the control of parasitic currents, for a variety of grid arrangements, and for different material properties. Several axisymmetric dam-break and rising bubble problems, for which highquality measured data are available, are also presented to serve as validation tests. In all cases, very good agreement of simulation results with experimental data has been recorded, and the dynamics of the problem well reproduced.