In this study, a novel methodology based on signal processing and machine learning approaches is proposed for real-time transient stability prediction (TSP) in power systems where the signals obtained from PMUs are utilized. The proposed method for TSP takes the computed energy of PMU signals in a window of measurements, as an input to a classifier to predict the stability of the system. Several types of classifiers, which are multi-layered perceptrons (MLPs), decision trees, and naive Bayes classifiers, are employed. Two alternative approaches of choosing the window of measurements used for TSP are proposed, where an MLP-based fault detection process is also proposed to form the proper window of measurements. One approach is to use a fixed window of only post fault measurements, whereas the other approach is to use an expanding window of measurements covering pre fault, fault-on and post-fault stages. Utilization of the energy concept in TSP gives the flexibility to process signals in different sizes while providing predictions robust to measurement noises and missing data. It also makes feature selection methods directly applicable, making the TSP possible with less PMUs. The proposed methods are applied to two different test systems and a large-scale model of the Turkish power system.