We present high average power picosecond and nanosecond pulse delivery at 1030 nm and 1064 nm wavelengths respectively through a novel hollow-core Negative Curvature Fiber (NCF) for high-precision micro-machining applications. Picosecond pulses with an average power above 36 W and energies of 92 µJ, corresponding to a peak power density of 1.5 TWcm⁻² have been transmitted through the fiber without introducing any damage to the input and output fiber end-faces. High-energy nanosecond pulses (>1 mJ), which are ideal for micro-machining have been successfully delivered through the NCF with a coupling efficiency of 92%. Picosecond and nanosecond pulse delivery have been demonstrated in fiber-based laser micro-machining of fused silica, aluminum and titanium.