In the past two decades, significant progress has been made in automatic speech recognition (ASR) and statistical machine translation (MT). Despite some conspicuous differences, many problems in ASR and MT are closely related and techniques in the two fields can be successfully cross-pollinated. In the September issue of IEEE Signal Processing Magazine, the Lecture Note column article explains the fundamental connections between ASR and MT. It shows that the recently developed ASR discriminative training paradigm can be extended to train MT models in the same spirit. Furthermore, speech translation (ST), which aims to translate speech sound from one language to another, usually consists of both ASR and MT as subcomponents. The column article shows that the ASR discriminative training paradigm can also be extended to ST.