Machine translation (MT) has been a challenging application of natural language processing technologies for many years. However, recent major improvements in translation accuracy have led to instances such as Web-based services that almost instantly translate any text into different languages or business-to-business services for high-quality translation of domain-specific documents. This article covers the foundation of recent MT systems and introduces translation as a mathematical process. It also focuses on how an MT system automatically learns to translate using samples of translated texts and how it renders output by combining acquired knowledge.