Heterogeneous manycore architectures are deployed to simultaneously run multiple and diverse applications. This requires various computing capabilities (CPUs, GPUs, and accelerators), and an efficient network-on-chip (NoC) architecture to concurrently handle diverse application communication behavior. However, supporting the concurrent communication requirements of diverse applications is challenging due to the dynamic application mapping, the complexity of handling distinct communication patterns and limited on-chip resources. In this paper, we propose Adapt-NoC, a versatile and flexible NoC architecture for chiplet-based manycore architectures, consisting of adaptable routers and links. Adapt-NoC can dynamically allocate disjoint regions of the NoC, called subNoCs, for concurrently-running applications, each of which can be optimized for different communication behavior. The adaptable routers and links are capable of providing various subNoC topologies, satisfying different latency and bandwidth requirements of various traffic patterns (e.g. all-to-all, one-to-many). Full system simulation shows that AdaptNoC can achieve 31% latency reduction, 24% energy saving and 10% execution time reduction on average, when compared to prior designs.