Recently, two-dimensional (2D) layered molybdenum disulfide (MoS2) has attracted great attention because of its graphene-like structure and unique physical and chemical properties. In this paper, physical structure, band gap structure, and optical properties of MoS2 are summarized. MoS2 is semiconducting and composed of covalently bonded sheets held together by weak van der Waals force. In each MoS2 layer, a layer of molybdenum (Mo) atoms is sandwiched between two layers of sulfur (S) atoms. There are three types of MoS2 compounds, including 1T MoS2, 2H MoS2, and 3R MoS2. As the number of layers decreases, the bad gap becomes larger. The bad gap transforms from indirect to direct as MoS2 is thinned to a monolayer. Changes of band gap show a great potential in photoelectron. Preparation methods of 2D MoS2 are reviewed, including growth methods and exfoliation methods. Ammonium thiomolybdate (NH4)2MoS4, elemental molybdenum Mo and molybdenum trioxide MoO3 are used to synthesize 2D MoS2 by growth methods. (NH4)2MoS4 is dissolved in a solution and then coated on a substrate. (NH4)2MoS4 is decomposed into MoS2 after annealing at a high temperature. Mo is evaporated onto a substrate, and then sulfurized into MoS2. MoO3 is most used to synthesize MoS2 on different substrates by a chemical vapor deposition or plasma-enhanced chemical vapor deposition. Other precursors like Mo(CO)6, MoS2 and MoCl5 are also used for MoS2 growth. For the graphene-like structure, monolayer MoS2 can be exfoliated from bulk MoS2. Exfoliation methods include micromechanical exfoliation, liquid exfoliation, lithium-based intercalation and electrochemistry lithium-based intercalation. For micromechanical exfoliation, the efficiency is low and the sizes of MoS2 flakes are small. For liquid exfoliation, it is convenient for operation to obtain mass production, but the concentration of monolayer MoS2 is low. For lithium-based intercalation, the yield of monolayer MoS2 is high while it takes a long time and makes 2H MoS2 transform to 1T MoS2 in this process. For electrochemistry lithium-based intercalation, this method saves more time and achieves higher monolayer MoS2 yield, and annealing makes 1T MoS2 back to 2H MoS2. The applications of 2D MoS2 in field-effect transistors, sensors and memory are discussed. On-off ratio field effect transistor based on MoS2 has field-effect mobility of several hundred cm2V-1-1 and on/off ratio of 108 theoretically.