Predicting human actions is a very actual research field. Artificial intelligence methods are commonly used here. They enable early recognition and classification of human activities. Such knowledge is extremely needed in the work on robots and other interactive systems that communicate and cooperate with people. This ensures early reactions of such devices and proper planning of their future actions. However, due to the complexity of human actions, predicting them is a difficult task. In this article, we review state-of-the-art methods and summarize recent advances in predicting human activity. We focus in particular on four approaches using machine learning methods, namely methods using: artificial neural networks, support vector machines, probabilistic models and decision trees. We discuss the advantages and disadvantages of these approaches, as well as current challenges related to predicting human activity. In addition, we describe the types of sensors and data sets commonly used in research on predicting and recognizing human actions. We analyze the quality of the methods used, based on the prediction accuracy reported in scientific articles. We describe the importance of the data type and the parameters of machine learning models. Finally, we summarize the latest research trends. The article is intended to help in choosing the right method of predicting human activity, along with an indication of the tools and resources necessary to effectively achieve this goal.