Greedy algorithms are ubiquitous in computational mathematics. In this talk, I will present novel convergence estimates of greedy algorithms including the reduced basis method for parametrized PDEs, the empirical interpolation method for approximating parametric functions, and the orthogonal/Chebyshev greedy algorithms for nonlinear dictionary approximation. The proposed convergence rates are all based on the metric entropy of underlying compact sets. This talk is partially based on joint works with Jonathan Siegel.