Nuclear molecules are analogous to ordinary electronic molecules. Valence nucleons are circling nuclear cores and thus bind them. They appear in collisions of nuclei on nuclei, and in fission and fusion processes. Here a lively field of research has developed over the past 20 years. Nuclear Molecules are the strongest deformed nuclear complexes and play an important role in nuclear structure (cluster) physics. They are also of considerable interest for the synthesis of elements in astrophysics (cosmology). Most of the various nuclear molecular phenomena are discussed. This book is the first monograph exclusively written to cover the theoretical aspects of nuclear molecular phenomena in heavy ion collisions. The experimental evidence is presented and confronted with theory. Contents: IntroductionA Brief Survey of Elementary Experimental DataTheoretical Bases for Nuclear MoleculesNucleus-Nucleus Interaction PotentialsExcitation Mechanisms and Models for Molecular Collective StatesFormation and Decay Widths of Nuclear Molecular ConfigurationsGeneral Theory of Heavy-Ion CollisionsApproximate Methods for Heavy-Ion CollisionsCollective Molecular States and Coupled Channels CalculationsReaction Theory in the Molecular Particle-Core ModelMolecular Single-Particle Configurations and the Nuclear Landau-Zenger EffectMicroscopic Molecular Reaction Theory in a Particle-Hole FormalismThree-Cluster Nuclear Molecules and Cluster Transfer in Heavy-Ion ReactionsFragmentation Theory and Collective Mass TransferSpeculative Molecular Effects Readership: Graduate students and active researchers interested in nuclear and heavy ion physics and nuclear astrophysics. keywords:Nuclear Molecules;Molecular Single-Particle Effects;Sub-Barrier Resonances;Elastic Scattering;Inelastic Scattering, Transfer Reactions;Total Reaction Cross Section;Fusion Reaction;Potential Resonances;y-Ray Measurements;;Sudden and Adiabatic Potentials;Optical Potential;Two-Center Shell Model;Liquid Drop Model;Shell Correction Method;Folding Model;Energy-Density Formalism;Proximity Model;Imaginary Potential;Double Resonance Mechanism;Band Crossing Model;Orbiting Cluster Model;Molecular Symmetry Model;Collective Two-Center Model;Cluster Model;Molecular Window;Coexistence Model;Shape Isomers in Fission;Atomic or One-Center Approach;Optical Model;Generator Coordinate Method;Rainbow Scattering;Glory Scattering;DWBA Formalism;Deep Inelastic Collisions;Nuclear Friction;Coupled Channel Calculations;Molecular Particle-Core Model;Molecular Wave Functions;