Capture-recapture methods are used to estimate the unknown size of a target population whose size cannot be reasonably enumerated. This thesis proposes the estimators and the models specifically designed to estimate the size of a population for one-inflated capture-recapture count data allowing for heterogeneity. These estimators can assist with overestimation problems occurring from one-inflation that can be seen in several areas of researches. The estimators are developed under three approaches. The first approach is based on a modification by truncating singletons and applying the conventional Turing and maximum likelihood estimation approach to the one-truncated geometric data for estimating the parameter p₀. These p₀ are applied to the Horvitz-Thompson approach for the modified Turing estimator (T_OT) and the modified maximum likelihood estimator (MLE_OT). The second approach is the model-based approach. It focuses on developing a statistical model that describes the mechanism to generate the extra of count ones. The new estimator MLE_ZTOI is developed from a maximum likelihood approach by using the nested EM algorithm based upon the zero-truncated one-inflated geometric distribution. The last approach focuses on modifying a classical Chao’s estimator to involve the frequency of counts of twos and threes instead of the frequency of counts of ones and twos. The modified Chao estimator (MC) is asymptotic unbiased estimator for a power series distribution with and without one-inflation and provides a lower bound estimator under a mixture of power series distributions with and without one-inflation. The three bias-correction versions of the modified Chao estimator have been developed to reduce the bias when the sample size is small. A variance approximation of MC and MC3 are also constructed by using a conditioning technique. All of the proposed estimators are assessed through simulation studies. The real data sets are provided for understanding the methodologies.