Feature selection is important in many big data applications. There are at least two critical challenges. Firstly, in many applications, the dimensionality is extremely high, in millions, and keeps growing. Secondly, feature selection has to be highly scalable, preferably in an online manner such that each feature can be processed in a sequential scan. In this paper, we develop SAOLA, a Scalable and Accurate OnLine Approach for feature selection. With a theoretical analysis on bounds of the pairwise correlations between features, SAOLA employs novel online pairwise comparison techniques to address the two challenges and maintain a parsimonious model over time in an online manner. Furthermore, to tackle the dimensionality that arrives by groups, we extend our SAOLA algorithm, and then propose a novel group-SAOLA algorithm for online group feature selection. The group-SAOLA algorithm can online maintain a set of feature groups that is sparse at the level of both groups and individual features simultaneously. An empirical study using a series of benchmark real data sets shows that our two algorithms, SAOLA and group-SAOLA, are scalable on data sets of extremely high dimensionality, and have superior performance over the state-of-the-art feature selection methods.