We report on the design and implementation of a field-programmable Josephson amplifier (FPJA)-a compact and lossless superconducting circuit that can be programmed in situ by a set of microwave drives to perform reciprocal and nonreciprocal frequency conversion and amplification. In this work, we demonstrate four modes of operation: frequency conversion (transmission of -0.5 dB, reflection of -30 dB), circulation (transmission of -0.5 dB, reflection of -30 dB, isolation of 30 dB), phase-preserving amplification (gain > 20 dB, one photon of added noise) and directional phase-preserving amplification (reflection of -10 dB, forward gain of 18 dB, reverse isolation of 8 dB, one photon of added noise). The system exhibits quantitative agreement with the theoretical prediction. Based on a gradiometric superconducting quantum-interference device with Josephson junctions, the FPJA is first-order insensitive to flux noise and can be operated without magnetic shielding at low temperature. Owing to its flexible design and compatibility with existing superconducting fabrication techniques, the FPJA offers a straightforward route toward on-chip integration with superconducting quantum circuits such as qubits and microwave optomechanical systems.