This paper reports the fabrication and characterization of high-quality exfoliated mono- and bilayer graphene (MLG and BLG) devices on single-crystal hexagonal boron nitride (h-BN) substrates using a mechanical transfer process. The study highlights the advantages of h-BN as a substrate compared to standard SiO$_2$, which often leads to disordered graphene devices with inferior properties. h-BN is an insulating material with a large band gap and a small lattice mismatch with graphite, making it chemically inert and free of dangling bonds or surface charge traps. The atomic planarity of h-BN suppresses rippling in graphene, enhancing device performance. Key findings include: 1. **Enhanced Mobility**: MLG and BLG devices on h-BN exhibit significantly higher carrier mobility compared to those on SiO$_2$. For MLG, the mobility is approximately three times higher, and for BLG, it is comparable to the best reported suspended BLG devices. 2. **Reduced Carrier Inhomogeneity**: The resistivity peak at the charge neutrality point (CNP) is narrower and occurs at nearly zero gate voltage, indicating reduced carrier inhomogeneity. 3. **Improved Doping**: h-BN supports less intrinsic doping, leading to a reduction in carrier inhomogeneity. 4. **Quantum Hall Effects**: Both MLG and BLG devices on h-BN exhibit quantized Hall conductance and Shubnikov-de Haas oscillations, confirming the high quality of the graphene. 5. **Annealing Effects**: Annealing in H$_2$/Ar gas enhances carrier mobility in MLG, while no significant doping occurs, unlike on SiO$_2$. The study demonstrates that h-BN substrates can significantly improve the quality and functionality of graphene devices, paving the way for advanced graphene electronics and heterostructures.This paper reports the fabrication and characterization of high-quality exfoliated mono- and bilayer graphene (MLG and BLG) devices on single-crystal hexagonal boron nitride (h-BN) substrates using a mechanical transfer process. The study highlights the advantages of h-BN as a substrate compared to standard SiO$_2$, which often leads to disordered graphene devices with inferior properties. h-BN is an insulating material with a large band gap and a small lattice mismatch with graphite, making it chemically inert and free of dangling bonds or surface charge traps. The atomic planarity of h-BN suppresses rippling in graphene, enhancing device performance. Key findings include: 1. **Enhanced Mobility**: MLG and BLG devices on h-BN exhibit significantly higher carrier mobility compared to those on SiO$_2$. For MLG, the mobility is approximately three times higher, and for BLG, it is comparable to the best reported suspended BLG devices. 2. **Reduced Carrier Inhomogeneity**: The resistivity peak at the charge neutrality point (CNP) is narrower and occurs at nearly zero gate voltage, indicating reduced carrier inhomogeneity. 3. **Improved Doping**: h-BN supports less intrinsic doping, leading to a reduction in carrier inhomogeneity. 4. **Quantum Hall Effects**: Both MLG and BLG devices on h-BN exhibit quantized Hall conductance and Shubnikov-de Haas oscillations, confirming the high quality of the graphene. 5. **Annealing Effects**: Annealing in H$_2$/Ar gas enhances carrier mobility in MLG, while no significant doping occurs, unlike on SiO$_2$. The study demonstrates that h-BN substrates can significantly improve the quality and functionality of graphene devices, paving the way for advanced graphene electronics and heterostructures.