Quantum compilation is the task of translating a high-level description of a quantum algorithm into a sequence of low-level quantum operations. We propose and motivate the use of Xor-And-Inverter Graphs (XAG) to specify Boolean functions for quantum compilation. We present three different XAG-based compilation algorithms to synthesize quantum circuits in the Clifford + T library, hence targeting fault-tolerant quantum computing. The algorithms are designed to minimize relevant cost functions, such as the number of qubits, the T-count, and the T-depth, while allowing the flexibility of exploring different solutions. We present novel resource estimation results for relevant cryptographic and arithmetic benchmarks. The achieved results show a significant reduction in both T-count and T-depth when compared with the state-of-the-art.