Multi-modal conversation emotion recognition (MCER) aims to recognize and track the speaker's emotional state using text, speech, and visual information in the conversation scene. Analyzing and studying MCER issues is significant to affective computing, intelligent recommendations, and human-computer interaction fields. Unlike the traditional single-utterance multi-modal emotion recognition or single-modal conversation emotion recognition, MCER is a more challenging problem that needs to deal with more complex emotional interaction relationships. The critical issue is learning consistency and complementary semantics for multi-modal feature fusion based on emotional interaction relationships. To solve this problem, people have conducted extensive research on MCER based on deep learning technology, but there is still a lack of systematic review of the modeling methods. Therefore, a timely and comprehensive overview of MCER's recent advances in deep learning is of great significance to academia and industry. In this survey, we provide a comprehensive overview of MCER modeling methods and roughly divide MCER methods into four categories, i.e., context-free modeling, sequential context modeling, speaker-differentiated modeling, and speaker-relationship modeling. In addition, we further discuss MCER's publicly available popular datasets, multi-modal feature extraction methods, application areas, existing challenges, and future development directions. We hope that our review can help MCER researchers understand the current research status in emotion recognition, provide some inspiration, and develop more efficient models.