We discuss in this paper the propagation of lightning current pulses along conical tall structures. Although the dominant mode of an infinitely long conical transmission line is transverse electric and magnetic (TEM), such a structure can also support higher order TE and TM modes which display a gradual cutoff frequency variation with height. Recently, Baba and Rakov's finite-difference time domain numerical analysis revealed that for a perfectly conducting conical structure, while the current pulses suffer no attenuation as they travel from the cone's apex to its base, the attenuation is significant when pulses propagate from the base to the apex. Adopting an analysis method using 1) the COMSOL Multiphysics simulation environment based on the finite element method and 2) the partial equivalent element circuit method, we study the same reduced-scale structure analyzed by Baba and Rakov. The obtained results confirm the conclusions drawn by Baba and Rakov. We also perform simulations for the case of a 100-m tall tower considering different tower-base radii. It is shown that the upward current pulses are affected by a strong attenuation resulting from the field scattering near the discontinuity at the tower base, followed by a weaker attenuation resulting from the propagation along the cone from its base to the apex. A simple way to modify the engineering lightning return stroke models to account for the attenuation of the upward current pulses is suggested. Finally, we report on experiments to study the current pulse propagation along a 1/582 reduced scale model of the Toronto, CN, tower. The obtained experimental data support the numerical simulations.