This paper develops a systematic approach to characterize 't Hooft anomalies in open quantum systems, where symmetries manifest as either strong or weak types due to nontrivial couplings to the environment. The authors represent these symmetries through superoperators and incorporate them into a unified framework to calculate anomalies. For systems with full symmetry groups $K \times G$, where $K$ is the strong symmetry and $G$ is the weak symmetry, anomalies of bosonic systems are classified by $H^{1+2}(K \times G, U(1))/H^{1+2}(G, U(1))$ in $d$ spatial dimensions. The paper demonstrates that anomalies lead to nontrivial mixed-state quantum phases, ensuring nontrivial steady states and long-time dynamics for open quantum systems governed by Lindbladians. A novel $(1+1)$-D mixed-state quantum phase is identified, characterized by spontaneous symmetry breaking on the boundary enforced by anomalies, with no counterpart in closed systems. The authors also explore the "anomaly inflow" mechanism in open quantum systems, constructing Lindbladians with mixed-state symmetry-protected topological order in the bulk and nontrivial edge theories.This paper develops a systematic approach to characterize 't Hooft anomalies in open quantum systems, where symmetries manifest as either strong or weak types due to nontrivial couplings to the environment. The authors represent these symmetries through superoperators and incorporate them into a unified framework to calculate anomalies. For systems with full symmetry groups $K \times G$, where $K$ is the strong symmetry and $G$ is the weak symmetry, anomalies of bosonic systems are classified by $H^{1+2}(K \times G, U(1))/H^{1+2}(G, U(1))$ in $d$ spatial dimensions. The paper demonstrates that anomalies lead to nontrivial mixed-state quantum phases, ensuring nontrivial steady states and long-time dynamics for open quantum systems governed by Lindbladians. A novel $(1+1)$-D mixed-state quantum phase is identified, characterized by spontaneous symmetry breaking on the boundary enforced by anomalies, with no counterpart in closed systems. The authors also explore the "anomaly inflow" mechanism in open quantum systems, constructing Lindbladians with mixed-state symmetry-protected topological order in the bulk and nontrivial edge theories.