This paper introduces INCPrompt, an innovative continual learning solution that effectively addresses catastrophic forgetting. INCPrompt's key innovation lies in its use of adaptive key-learner and task-aware prompts that capture task-relevant information. This unique combination encapsulates general knowledge across tasks and encodes task-specific knowledge. Our comprehensive evaluation across multiple continual learning benchmarks demonstrates INCPrompt's superiority over existing algorithms, showing its effectiveness in mitigating catastrophic forgetting while maintaining high performance. These results highlight the significant impact of task-aware incremental prompting on continual learning performance.