Abstract      In this paper, a new family of discrete alpha power distributions is introduced. Some properties including quantiles, mean residual life, mean time to failure, R nyi entropy, moments and order statistics are obtained. Discrete alpha power Weibull distribution, as a member from this family, is studied in detail. Discrete two-parameter Weibull distribution, discrete alpha power one parameter Weibull distribution, discrete alpha power exponential distribution, discrete one parameter Weibull distribution, discrete Rayleigh distribution, discrete exponential distribution, discrete alpha power Rayleigh distribution are sub models of discrete alpha power Weibull distribution. A simulation study is conducted to investigate the precision of the theoretical results based on simulated and real data through some measurements of accuracy. Three real data sets are analyzed to illustrate the suitability and applicability of the proposed model.