Count regression models maintain a steadfast presence in modern applied statistics as highlighted by their usage in diverse areas like biometry, ecology, and insurance. However, a common practical problem with observed count data is the presence of excess zeros relative to the assumed count distribution. The seminal work of Lambert (1992) was one of the first articles to thoroughly treat the problem of zero-inflated count data in the presence of covariates. Since then, a vast literature has emerged regarding zero-inflated count regression models. In this first of two review articles, we survey some of the classic and contemporary literature on parametric zero-inflated count regression models, with emphasis on the utility of different univariate discrete distributions. We highlight some of the primary computational tools available for estimating and assessing the adequacy of these models. We concurrently emphasize the diverse data problems to which these models have been applied.