Task-based programming models have become very popular, as they offer an attractive solution to parallelize serial application code with task and data annotations. They usually depend on a runtime system that schedules the tasks to multiple cores in parallel while resolving any data hazards. However, existing runtime system implementations are not ready to scale well on emerging manycore processors, as they often rely on centralized structures and/or locks on shared structures in a cache-coherent memory. We propose design choices, policies and mechanisms to enhance runtime system scalability for single-chip processors with hundreds of cores. Based on these concepts, we create and evaluate Myrmics, a runtime system for a dependency-aware, task-based programming model on a heterogeneous hardware prototype platform that emulates a single-chip processor of 8 latency-optimized and 512 throughput-optimized CPUs. We find that Myrmics scales successfully to hundreds of cores. Compared to MPI versions of the same benchmarks with hand-tuned message passing, Myrmics achieves similar scalability with a 10-30% performance overhead, but with less programming effort. We analyze the scalability of the runtime system in detail and identify the key factors that contribute to it.