This chapter is a largely self-contained introduction to nearly linear-time, practical algorithms for submodular maximization, for both monotone and non-monotone objective functions, culminating in recent results achieving nearly the optimal ratio in linear time. The algorithms include various fast, greedy algorithms for monotone functions, and double and simultaneous greedy approaches for non-monotone functions. We focus the presentation on the size constraint, since the algorithms for size constraint typically generalize, sometimes directly, to more sophisticated constraint systems. In addition to the main ideas, we provide detailed theoretical analysis for most of the algorithms.