The Khintchine inequality has played a crucial role in various theories of analysis and, surprisingly, there is a natural and successful analogue in the setting of non-commutative probability spaces. In this talk, I will explain what the non-commutative Khintchine inequality is and how it recovers a theorem of Littlewood on random Fourier series. Also, I will address its implication on the study of Fourier multipliers from L^p into l^1.