Resilience, which is defined as the ability of a system to adjust its activity for retaining the basic functionality when errors, failures and environmental changes occur, is an essential and fundamental property for biomolecular networks. The studies of biomolecular network resilience attempt to answer the following 3 questions. (1) What is the potential mechanism of the resilience of biomolecular networks? (2) How the state of biomolecular network migrates from one stable steady state to another under the effect of resilience? (3) How to predict the tipping points of state transitions to prevent the system from evolving into undesirable states (such as disease states)? In view of the importance of resilience for biomolecular networks and its clinical application value, we systematically review the research progress focusing on 3 questions above in the past 20 years. As one of the important steady-state characteristics of resilience systems, bi-stability (or multi-stability) can help us to uncover the underlying mechanism of the resilience. Biomolecular networks consist of numerous repeated network motifs, and the steady-state response of almost all network motifs which contain feedback loops (e.g. positive autoregulation motif, mutual inhibition motif) is bi-stability (or multi-stability). Based on our numeric simulation, the network motifs with feedback loops have different steady-state response characteristics although they all bi-stability (or multi-stability), which result in the different biological functions they can describe. Many studies also indicated that stochastic noise from internal or external could affect the number of stable-steady states and hysteresis of the network motifs with bi-stability (or multi-stability). Furthermore, the bi-stable network motifs have been used to model many biological processes, such as cell cycle and embryonic development, to reveal their mechanism. The network motifs are too simple to model complex biological processes, which generally involved in interactions between lots of biomolecules. Potential function, as a powerful tool in the field of dynamical system, is widely used to uncover the state transition properties of high-dimensional biomolecular networks. Many methods have been proposed to reconstruct the potential function of equilibrium systems and non-equilibrium systems based on network dynamics. Using these methods, a vast number of studies revealed the state transition mechanism of various biological processes, such as cell differentiation, cancer initiation, etc. The state of biomolecular network generally migrates from one stable steady state to another abruptly and drastically under the effect of resilience. However, detecting tipping points of state transitions in system level is impossible based on network dynamic, due to the complexity and nonlinearity of biomolecular networks. To fill this gap, many indicators have been proposed to predict the upcoming tipping points from biological data under network perspective, such as dynamic network biomarker (DNB). And these indicators were also used to detect the critical transitions in the development of various diseases (e.g., diabetes, influenza, cancer). So far, the study of biomolecular network resilience has been helped us to understand the mechanism of state transitions in many biological processes. However, there are still some important issues that have not been resolved. (1) Studying the resilience of large-scale biomolecular networks which consist of thousands of biomolecules; (2) reconstructing potential function of large-scale biomolecular networks based on biological data; (3) control biomolecular network resilience.