This will be a survey talk about recent progress on norm and pointwise convergence problems for classical and multiple ergodic averages along polynomial orbits. A celebrated theorem of Szemeredi asserts that every subset of integers with nonvanishing upper Banach density contains arbitrarily long arithmetic progressions. We will discuss the significance of using ergodic theory and Fourier analysis in solving this problem. We will also explain how this problem led to the conjecture of Furstenberg-Bergelson-Leibman, which is a major open problem in pointwise ergodic theory. Relations with number theory and additive combinatorics will be also discussed.