Tourism demand is the basis on which all commercial decisions concerning tourism ultimately depend. Accurate estimation of tourism demand is essential for the tourism industry because it can help reduce risk and uncertainty as well as effectively provide basic information for better tourism planning. The purpose of this study is to develop the optimal forecasting model that yields the highest accuracy when compared to the forecast performances of three different methods, namely Artificial Neural Network (ANN), Exponential Smoothing, and Box-Jenkins methods for forecasting monthly inbound tourist flows to Croatia. Prior studies have been applied to forecast tourism demand to Croatia based on time series models and casual methods. However, the monthly and comparative tourism demand forecasting studies using ANNs are still limited, and this paper aims to fill this gap. The number of monthly foreign tourist arrivals to Croatia covers the period between January 2005-December 2019 data were used to build optimal forecasting models. Forecasting performances of the models were measured by Mean Absolute Percentage Error (MAPE) statistics. As a result of the experiments carried out, when compared to the forecasting performances of various models, 12 lagged ANN models, which have [4-3-1] architecture, were seen to perform best among all models applied in this study. Considering both the empirical findings obtained from this study and previous studies on tourism forecasting, it can be seen that ANN models that do not have any negativities (such as over-training, faulty architecture, etc.) produce successful forecasting results when compared with results generated by conventional statistical methods.