People usually have the explicit or implicit desire to get the information they need and are most interested in from massive information, which has led to the creation of personalized recommender systems. Recommender systems are set up to address the issue of information overload in traditional information retrieval systems such as search engines, and have been a significant area of research focused on recommending information that is of most interest to users. There is a sequential nature to the behavior of a person interacting with a system, such as examining one item of clothing before examining others. The problem of taking this sequential nature into account in delivering recommendation is known as sequential recommendation (SR). The traditional sequential recommendation problem merely takes into account a single type of behavior of the users, while in real scenarios users tend to engage in multiple types of behaviors, such as examining and adding clothes to cart before purchasing them, leading to the proposal of multibehavior sequential recommendation (MBSR). MBSR considers both sequentiality and heterogeneity of user behaviors, which can achieve state-of-the-art recommendation through suitable modeling. Hence, MBSR is a relatively new and worthy direction for in-depth research, for which some related works have been proposed. This survey aims to shed light on the MBSR problem. Firstly, we introduce MBSR in detail, including its problem definition, application scenarios and challenges faced. Secondly, we detail the classification of MBSR, including neighborhood-based methods, matrix factorization-based methods and deep learningbased methods, where we further classify the deep learningbased methods into different learning architectures based on RNN, GNN, Transformer, and generic architectures as well as architectures that integrate hybrid techniques. In each method, we present related works based on the data perspective and the modeling perspective, as well as analyze the strengths, weaknesses and features of these works. Finally, we discuss some promising future research directions to address the challenges and improve the current status of MBSR.