We study a fractional birth-death process with state dependent birth and death rates. It is defined using a system of fractional differential equations that generalizes the classical birth-death process introduced by Feller (1939). We obtain the closed form expressions for its transient probabilities using Adomian decomposition method. In this way, we obtain the unknown transient probabilities for the classical birth-death process (see Feller (1968), p. 454). Its various distributional properties are studied. For the case of linear birth and death rates, the obtained results are verified with the existing results. Also, we discuss the cumulative births in the fractional linear birth-death process. Later, we consider a time-changed linear birth-death process where we discuss the asymptotic behaviour of the distribution function of its extinction time at zero.