In this work, we present a family of distributions for count data, named Zero-Modified Power Series (ZMPS), an extension of the Power Series distributions family whose support starts at zero. This extension consists of modifying the probability of observing zero of each Power Series distribution, allowing the new zero-modified distribution appropriately accommodate datasets that have any amount of zero observations (for instance, zero-inflated or zero-deflated datasets). Power Series distributions included in the Zero-Modified Power Series family are Poisson, Generalized Poisson, Geometric, Binomial, Negative Binomial, and Generalized Negative Binomial. In addition, we introduce the Zero-Modified Power Series regression models and propose a Bayesian approach. Two real datasets are analyzed: the first corresponds to leptospirosis notifications in cities of Bahia State in Brazil; the second corresponds to the number of goals scored by a team in a sports competition.