We present a comprehensive analysis of the morphology and dynamics of relativistic pressure-matched axisymmetric jets. The numerical simulations have been carried out with a high-resolution shock-capturing hydrocode based on an approximate relativistic Riemann solver derived from the spectral decomposition of the Jacobian matrices of relativistic hydrodynamics. We discuss the dependence of the jet morphology on several parameters, paying special attention to the relativistic effects caused by high Lorentz factors and large internal energies of the beam flow. The parameter space of our analysis is spanned by the ratio of the beam and ambient medium rest mass density (η), the beam Mach number (Mb), the beam Lorentz factor (Wb), and the adiabatic index (γ) of the equation of state (assuming an ideal gas). Both the ultrarelativistic regime (Wb >= 20) and the hypersonic regime (relativistic Mach number greater than 100) have been studied.Our results show that the enhancement of the effective inertial mass of the beam due to relativistic effects (through the specific enthalpy and the Lorentz factor) makes relativistic jets significantly more stable than Newtonian jets. We find that relativistic jets propagate very efficiently through the ambient medium, at speeds that agree very well with those obtained from an estimate based on a one-dimensional momentum balance. The propagation efficiency of a relativistic jet is an increasing function of the beam flow velocity. Relativistic jets seem to give rise to two different morphologies, according to the relevance of relativistic effects. Hot beams (i.e., with internal energies comparable to the beam rest-mass energy) show little internal structure (as they are almost in pressure equilibrium with their surroundings) and relatively smooth cocoons forming lobes near the head of the jet. Highly supersonic models, in which the kinematic relativistic effects due to high beam flow Lorentz factors dominate, display extended cocoons that are overpressured with respect to the environment. The cocoon thickness decreases, and its mean pressure increases with increasing beam Lorentz factor.