Count time series data are observed in several applied disciplines such as in environmental science, biostatistics, economics, public health, and finance. In some cases, a specific count, say zero, may occur more often than usual. Additionally, serial dependence might be found among these counts if they are recorded over time. Overlooking the frequent occurrence of zeros and the serial dependence could lead to false inference. In this dissertation, we propose two classes of copula-based time series models for zero-inflated counts with the presence of covariates. Zero-inflated Poisson (ZIP), zero-inflated negative binomial (ZINB), and zero-inflated Conway-Maxwell-Poisson (ZICMP) distributed marginals of the counts will be considered. For the first class, the joint distribution is modeled under Gaussian copula with autoregression moving average (ARMA) errors. Relationship between the autocorrelation function of the zero-inflated counts and the errors is studied. Sequential sampling likelihood inference is performed. To evaluate the proposed method, simulated and real-life data examples are provided and studied. For the second class, Markov zero-inflated count time series models based on a joint distribution on consecutive observations are proposed. The joint distribution function of the consecutive observations is constructed through copula functions.First or second order Markov chains are considered with the univariate margins of ZIP, ZINB, or ZICMP distributions. Under the Markov models, bivariate copula functions such as the bivariate Gaussian, Frank, and Gumbel are chosen to construct a bivariate distribution of two consecutive observations. Moreover, the trivariate Gaussian and max-infinitely divisible copula functions are considered to build the joint distribution of three consecutive observations. Likelihood based inference is performed, score functions are derived, and asymptotic properties are studied. Model diagnostic and prediction are presented. To evaluate the proposed method, simulated and real-life data examples are studied.