Quantum state engineering with Josephson-junction devices involves controlling the coherent dynamics of quantum systems, particularly qubits, using Josephson junctions. This review discusses the quantum properties of low-capacitance Josephson junction devices, focusing on charge and flux-based qubits. Charge qubits are based on Cooper pair charges on small islands or flux in ring geometries near degeneracy points. These qubits can be manipulated using gate voltages or magnetic fields, and their quantum states can be read out via coupling to a single-electron transistor (SET). Flux qubits, based on the phase of a Josephson junction or flux in a ring, also show promise for quantum computing. The review highlights the importance of long phase coherence times and minimizing dephasing effects. It discusses the quantum measurement process, including the role of the density matrix and master equation, and the challenges of maintaining coherence during measurements. The review also covers the experimental verification of quantum superpositions, coherent oscillations, and entangled states in Josephson qubits. The potential applications of these devices in quantum computing and logic operations are emphasized, along with the challenges of scaling up to large numbers of qubits and minimizing decoherence. The review concludes with the importance of further experiments to realize quantum logic elements and the need for continued research in quantum state engineering.Quantum state engineering with Josephson-junction devices involves controlling the coherent dynamics of quantum systems, particularly qubits, using Josephson junctions. This review discusses the quantum properties of low-capacitance Josephson junction devices, focusing on charge and flux-based qubits. Charge qubits are based on Cooper pair charges on small islands or flux in ring geometries near degeneracy points. These qubits can be manipulated using gate voltages or magnetic fields, and their quantum states can be read out via coupling to a single-electron transistor (SET). Flux qubits, based on the phase of a Josephson junction or flux in a ring, also show promise for quantum computing. The review highlights the importance of long phase coherence times and minimizing dephasing effects. It discusses the quantum measurement process, including the role of the density matrix and master equation, and the challenges of maintaining coherence during measurements. The review also covers the experimental verification of quantum superpositions, coherent oscillations, and entangled states in Josephson qubits. The potential applications of these devices in quantum computing and logic operations are emphasized, along with the challenges of scaling up to large numbers of qubits and minimizing decoherence. The review concludes with the importance of further experiments to realize quantum logic elements and the need for continued research in quantum state engineering.