Patients in an Intensive Care Unit (ICU) are closely and continuously monitored, and many machine learning (ML) solutions have been proposed to predict specific outcomes like death, bleeding, or organ failure. Forecasting of vital parameters is a more general approach to ML-based patient monitoring, but the literature on its feasibility and robust benchmarks of achievable accuracy are scarce. We implemented five univariate statistical models (the naïve model, the Theta method, exponential smoothing, the autoregressive integrated moving average model, and an autoregressive single-layer neural network), two univariate neural networks (N-BEATS and N-HiTS), and two multivariate neural networks designed for sequential data (a recurrent neural network with gated recurrent unit, GRU, and a Transformer network) to produce forecasts for six vital parameters recorded at five-minute intervals during intensive care monitoring. Vital parameters were the diastolic, systolic, and mean arterial blood pressure, central venous pressure, peripheral oxygen saturation (measured by non-invasive pulse oximetry) and heart rate, and forecasts were made for 5 through 120 minutes into the future. Patients used in this study recovered from cardiothoracic surgery in an ICU. The patient cohort used for model development (n = 22,348) and internal testing (n = 2,483) originated from a heart center in Germany, while a patient sub-set from the eICU collaborative research database, an American multicenter ICU cohort, was used for external testing (n = 7,477). The GRU was the predominant method in this study. Uni- and multivariate neural network models proved to be superior to univariate statistical models across vital parameters and forecast horizons, and their advantage steadily became more pronounced for increasing forecast horizons. With this study, we established an extensive set of benchmarks for forecast performance in the ICU. Our findings suggest that supplying physicians with short-term forecasts of vital parameters in the ICU is feasible, and that multivariate neural networks are most suited for the task due to their ability to learn patterns across thousands of patients.