Sarcasm is a complex form of communication where intended meanings are sometimes opposite to the literal ones. The challenge of automatic detection of sarcasm is doubled in low-resource languages such as Hindi and code-mixed text. In code-mixed languages, speakers mix multiple languages in a single discourse. Code-mixed Indic languages such as Hinglish (Hindi + English) are rampant in social media and other platforms. It involves an extra level of linguistic complexity in sarcasm detection by understanding and interpreting cultural nuances. This review paper covers many approaches, from classical ML models to modern transformer-based architectures, datasets, embedding techniques, and evaluation metrics used in previous studies to detect sarcasm in Hindi and Hinglish text. This paper provides the key challenges of the lack of annotated datasets, the complexity of the language mixing, and the role of contextual cues within a mechanism for detecting sarcasm. Thus, it identifies the research gaps and suggests future directions for the researchers.