Recoil-ion and electron momentum spectroscopy, using Reaction Microscopes, is a rapidly developing technique for measuring the vector momenta of ions and electrons from atomic or molecular fragmentation. These microscopes, derived from recoil-ion and COLTRIMS, offer large solid angles (up to 4π) and high momentum resolution (a few percent of an atomic unit). They enable the study of many-particle quantum dynamics in atomic and molecular systems, surfaces, and solids under time-dependent electromagnetic fields. This review focuses on recent technical developments and four new fragmentation experiments. Multi-dimensional momentum space images provide insights into single-photon induced molecular fragmentation and structure. Breakthroughs in high-intensity laser-induced fragmentation of atoms and molecules have been achieved. Two-electron reactions have matured to report fully differential cross sections (FDCS). Comprehensive FDCS for single ionization of atoms by ion impact have provided new insights and challenges to theory. The review also includes a brief summary of kinematics and envisions the method's future potential. The review discusses the kinematics of atomic fragmentation processes induced by electron, ion, photon, or laser-pulse impact. It emphasizes the role of the recoiling target ion and its momentum. The collision kinematics is fully determined by measuring 3N-3 linear independent momentum components in kinematically complete experiments. FDCS can be extracted from these measurements. Fast ion-atom collisions are analyzed, with momentum balances decoupled into longitudinal and transverse components. The recoil-ion momentum provides information on projectile deflection and reaction inelasticity. For photon collisions, the recoil-ion and electron momenta are opposite, compensating each other. The recoil-ion momentum distribution shows circular structures, with inner circles due to electron excitation. Imaging techniques, including Reaction Microscopes, enable high-resolution detection of recoil ions and electrons. These devices use position-sensitive detectors and magnetic fields to project momentum information. Electrons are imaged using magnetic fields to confine their trajectories, allowing precise momentum reconstruction. The transverse momentum of electrons is determined from their detection position and time-of-flight. Target preparation involves supersonic gas-jets and magneto-optical traps (MOTRIMS). Supersonic gas-jets produce cold, localized atomic or molecular beams, while MOTRIMS use laser-cooled atoms in magnetic traps. These methods reduce target temperature and enable high-resolution momentum spectroscopy. New developments include improvements in detector technology, such as delay-line anodes with negligible dead-time and pixel anodes with ultra-fast readout. The flexibility of imaging spectrometers can be increased by using time-dependent fields for specific applications. Pulsed extraction fields enhance detection efficiency for molecular fragmentation. These advancements continue to expand the capabilities of Reaction Microscopes in studying atomic and molecular reactions.Recoil-ion and electron momentum spectroscopy, using Reaction Microscopes, is a rapidly developing technique for measuring the vector momenta of ions and electrons from atomic or molecular fragmentation. These microscopes, derived from recoil-ion and COLTRIMS, offer large solid angles (up to 4π) and high momentum resolution (a few percent of an atomic unit). They enable the study of many-particle quantum dynamics in atomic and molecular systems, surfaces, and solids under time-dependent electromagnetic fields. This review focuses on recent technical developments and four new fragmentation experiments. Multi-dimensional momentum space images provide insights into single-photon induced molecular fragmentation and structure. Breakthroughs in high-intensity laser-induced fragmentation of atoms and molecules have been achieved. Two-electron reactions have matured to report fully differential cross sections (FDCS). Comprehensive FDCS for single ionization of atoms by ion impact have provided new insights and challenges to theory. The review also includes a brief summary of kinematics and envisions the method's future potential. The review discusses the kinematics of atomic fragmentation processes induced by electron, ion, photon, or laser-pulse impact. It emphasizes the role of the recoiling target ion and its momentum. The collision kinematics is fully determined by measuring 3N-3 linear independent momentum components in kinematically complete experiments. FDCS can be extracted from these measurements. Fast ion-atom collisions are analyzed, with momentum balances decoupled into longitudinal and transverse components. The recoil-ion momentum provides information on projectile deflection and reaction inelasticity. For photon collisions, the recoil-ion and electron momenta are opposite, compensating each other. The recoil-ion momentum distribution shows circular structures, with inner circles due to electron excitation. Imaging techniques, including Reaction Microscopes, enable high-resolution detection of recoil ions and electrons. These devices use position-sensitive detectors and magnetic fields to project momentum information. Electrons are imaged using magnetic fields to confine their trajectories, allowing precise momentum reconstruction. The transverse momentum of electrons is determined from their detection position and time-of-flight. Target preparation involves supersonic gas-jets and magneto-optical traps (MOTRIMS). Supersonic gas-jets produce cold, localized atomic or molecular beams, while MOTRIMS use laser-cooled atoms in magnetic traps. These methods reduce target temperature and enable high-resolution momentum spectroscopy. New developments include improvements in detector technology, such as delay-line anodes with negligible dead-time and pixel anodes with ultra-fast readout. The flexibility of imaging spectrometers can be increased by using time-dependent fields for specific applications. Pulsed extraction fields enhance detection efficiency for molecular fragmentation. These advancements continue to expand the capabilities of Reaction Microscopes in studying atomic and molecular reactions.