Fast detection of dc faults in medium-voltage dc (MVDC) microgrids poses a challenge as such faults can cause severe damage to voltage-sourced converters within few milliseconds. This paper proposes a new traveling-wave (TW)-based method to detect, classify, and locate different dc fault types in MVDC microgrids. Unlike the existing TW-based protection and fault location methods, the proposed technique: (1) utilizes only the first locally measured TW after the inception of a fault and (2) focuses on the waveshape properties and polarity of the TW, rather than its arrival time. Therefore, the proposed method is faster than the existing techniques, and also requires no form of communication. As a result, it can effectively operate as both primary and backup protection. The proposed method is robust against high-resistance faults, and has been tested for fault resistances of up to 200 Ω. The performance of the proposed scheme has been assessed using a ±2.5 kV TN-S grounded MVDC microgrid under various conditions. The results verify the scheme's ultra-high-speed, accuracy, sensitivity, selectivity, and independence from system configuration. The concepts discussed in this paper are independent of the voltage level. Thus, the proposed method can be applied to other types of dc grids as well.