Modern datacenter networks provide very high capacity via redundant Clos topologies and low switch latency, but transport protocols rarely deliver matching performance. We present NDP, a novel data-center transport architecture that achieves near-optimal completion times for short transfers and high flow throughput in a wide range of scenarios, including incast. NDP switch buffers are very shallow and when they fill the switches trim packets to headers and priority forward the headers. This gives receivers a full view of instantaneous demand from all senders, and is the basis for our novel, high-performance, multipath-aware transport protocol that can deal gracefully with massive incast events and prioritize traffic from different senders on RTT timescales. We implemented NDP in Linux hosts with DPDK, in a software switch, in a NetFPGA-based hardware switch, and in P4. We evaluate NDP's performance in our implementations and in large-scale simulations, simultaneously demonstrating support for very low-latency and high throughput.