In this technological era, smart and intelligent systems that are integrated with artificial intelligence (AI) techniques, algorithms, tools, and technologies, have impact on various aspects of our daily life. Communication and interaction between human and machine using speech becomes increasingly important, since it is an obvious substitute for keyboards and screens in the communication process. Therefore, numerous technologies take advantage of speech such as Automatic Speech Recognition (ASR), where human natural speech for many languages is used as the mean to interact with machines. Majority of the related works for ASR concentrate on the development and evaluation of ASR systems that serve a single language (monolingual) only, such as Arabic, English, Chinese, French, and many others. However, research attempts that combine multiple languages (bilingual and multilingual) during the development and evaluation of ASR systems are very limited. This paper aims to provide comprehensive research background and fundamentals of bilingual ASR, and related works that have combined two languages for ASR tasks from 2010 through 2021. It also formulates research taxonomy and discusses open challenges to bilingual ASR research. Based on our literature investigation, it is clear that bilingual ASR using deep learning approach is highly demanded and is able to provide acceptable performance. In addition, many combinations of two languages such as Arabic-English, Arabic-Malay, and others, are not attempted yet by the research community, which can open new research opportunities. Finally, it is clear that ASR research is moving towards not only bilingual ASR, but also multilingual ASR.