Native crystalline cellulose is notoriously difficult to dissolve due to its dense hydrogen bond network between chains and weaker hydrophobic forces between cellulose sheets. N-Methylmorpholine N-oxide (NMMO), the solvent behind the Lyocell process, is one of the most successful commercial solvents for the nonderivatized dissolution of cellulose. In this process, water plays a very important role. Its presence at low concentrations allows NMMO to dissolve substantial amounts of cellulose, while at much higher concentrations it precipitates the crystalline fibers. Using all-atom molecular dynamics, we study the thermodynamic and structural properties of ternary solutions of cellulose, NMMO, and water. Using the two-phase thermodynamic method to calculate solvent entropy, we estimate the free energy of dissolution of cellulose as a function of the water concentration and find a transition of spontaneity that is in excellent agreement with experiment. In pure water, we find that cellulose dissolution is nonspontaneous, a result that is due entirely to strong decreases in water entropy. Although the combined effect of enthalpy on dissolution in water is negligible, we observe a net loss of hydrogen bonds, resulting in a change in hydrogen bond energy that opposes dissolution. At lower water concentrations, cellulose dissolution is spontaneous and largely driven by decreases in enthalpy, with solvent entropy playing only a very minor role. When searching for the root causes of this enthalpy decrease, a complex picture emerges in which not one but many different factors contribute to NMMO's good solvent behavior. The reduction in enthalpy is led by the formation of strong hydrogen bonds between cellulose and NMMO's N-oxide, intensified through van der Waals interactions between NMMO's nonpolar body and the nonpolar surfaces of cellulose and unhindered by water at low concentrations due to the formation of efficient hydrogen bonds between water and cellulose.