Recently, chemical vapor deposition (CVD) on copper has been becoming a main method for preparing large-area and highquality monolayer graphene. In this paper, we first briefly introduce the preliminary understanding of the microstructure and growth behavior of graphene on copper, and then focus on the recent progress on the quality improvement, number of layers control and transfer-free growth of graphene. In the end, we attempt to analyze the possible development of CVD growth of graphene in future, including the controlled growth of large-size single-crystal graphene and bilayer graphene with different stacking orders. Graphene, a novel two-dimensional crystal, holds great promise in a wide range of applications such as electronics, transparent electrodes, energy storage, and functional composites due to its excellent properties such as giant carrier mobility, superior thermal conductivity, high transparency and good chemical stability [1,2]. Among all the developed synthesis methods, chemical vapor deposition (CVD) has been attracting increased interests for growing large-area highquality graphene film. In 2009, Li et al.[3] first realized the CVD growth of centimeter-sized graphene films dominated by monolayers (about 95%) by using polycrystalline copper (Cu) foil as a substrate. These graphene films show high quality with a carrier mobility up to 4050 cm 2 V 1 s 1 . In contrast to the CVD growth of graphene on nickel (Ni), which follows a carbon segregation/precipitation mechanism, graphene grown on Cu is superior in both the controllability and uniformity of the number of layers due to the self-limiting surface adsorption growth mechanism [4]. Moreover, it has a great potential to grow very large films.For example, Bae et al. [5] have realized the roll-to-toll production of 30 inch monolayer predominated graphene films. However, there is a large variation in the quality of graphene on Cu prepared by different groups with the carrier mobility varying from several hundreds to several thousands of cm 2 V 1 s 1 . These values are far below the theoretical limit (~10 6 cm 2 V 1 s 1 ), and merely comparable to those of the commercial silicon materials. Meanwhile, it is necessary to develop the CVD process for growing highquality few-and multi-layer graphene since monolayer graphene is unable to satisfy the requirements of certain applications. In addition, it still remains a great challenge to realize the nondestructive and efficient transfer of large-area high-quality CVD grown graphene. Extensive efforts have been devoted to addressing the above issues in the past two years. In this paper, we will first briefly introduce the preliminary understanding of the microstructure and growth behavior of graphene on Cu, and then focus on the recent progress on the quality improvement, number of layers control and transfer-free growth of graphene. In the end, we attempt to analyze the possible development of CVD growth