We will first provide a brief overview of Brain-Computer Interface from a machine learning and signal processing perspective. In particular showing the wealth, the complexity and the difficulties of the data available, a truly enormous challenge: In real-time a multi-variate very strongly noise contaminated data stream is to be processed and neuroelectric activities are to be accurately differentiated. We will then in detail discuss the components of the data analysis chain employed in modern BCI systems, spanning all aspects from preprocessing and feature extraction, adaptive vs. fixed classification and feedback design.