Alfvén waves are low-frequency transverse waves propagating in a magnetized plasma. We define the Alfvén frequency ω 0 as ω 0 k V A cos θ, where k is the wave number, V A is the Alfvén speed, and θ is the angle between the wave vector and the ambient magnetic field. There are partially ionized plasmas in laboratory, space, and astrophysical plasma systems, such as in the solar chromosphere, interstellar clouds, and the earth ionosphere. The presence of neutral particles may modify the wave frequency and cause damping of Alfvén waves. The effects on Alfvén waves depend on two parameters: (1) α n n / n i, the ratio of neutral density (n n), and ion density (n i); (2) β ν n i / ω 0, the ratio of neutral collisional frequency by ions ν n i to the Alfvén frequency ω 0. Most of the previous studies examined only the limiting case with a relatively large neutral collisional frequency or β 1. In the present paper, the dispersion relation for Alfvén waves is solved for all values of α and β. Approximate solutions in the limit β 1 as well as β 1 are obtained. It is found for the first time that there is a forbidden zone (FZ) in the α - β parameter space, where the real frequency of Alfvén waves becomes zero. We also solve the wavenumber k from the dispersion equation for a fixed frequency and find the existence of a heavy damping zone (HDZ). We then examine the presence of FZ and HDZ for Alfvén waves in the ionosphere and in the solar chromosphere.