Feature selection is an important technique for data mining before a machine learning algorithm is applied. Despite its importance, most studies of feature selection are restricted to batch learning. Unlike traditional batch learning methods, online learning represents a promising family of efficient and scalable machine learning algorithms for large-scale applications. Most existing studies of online learning require accessing all the attributes/features of training instances. Such a classical setting is not always appropriate for real-world applications when data instances are of high dimensionality or it is expensive to acquire the full set of attributes/features. To address this limitation, we investigate the problem of Online Feature Selection (OFS) in which an online learner is only allowed to maintain a classifier involved only a small and fixed number of features. The key challenge of Online Feature Selection is how to make accurate prediction using a small and fixed number of active features. This is in contrast to the classical setup of online learning where all the features can be used for prediction. We attempt to tackle this challenge by studying sparsity regularization and truncation techniques. Specifically, this article addresses two different tasks of online feature selection: (1) learning with full input where an learner is allowed to access all the features to decide the subset of active features, and (2) learning with partial input where only a limited number of features is allowed to be accessed for each instance by the learner. We present novel algorithms to solve each of the two problems and give their performance analysis. We evaluate the performance of the proposed algorithms for online feature selection on several public datasets, and demonstrate their applications to real-world problems including image classification in computer vision and microarray gene expression analysis in bioinformatics. The encouraging results of our experiments validate the efficacy and efficiency of the proposed techniques.