The objective of this paper is to present exact analytical solutions for the torsional vibration of rods with non-uniform cross-section.Using appropriate transformations the equation of motion of torsional vibration of a rod with varying cross-section is reduced to analytically solvable standard differential equations whose form depends upon the specific area variation.Solutions are obtained for a rod with for a polynomial area variation.The solutions are obtained in terms of special functions such as Bessel and Neumann functions.Simple formulas to predict the natural frequencies of non-uniform rods with various end conditions are presented.The natural frequencies of variable cross-section rods for these end conditions are calculated and their dependence on taper is discussed.