Although copper oxide high-temperature superconductors constitute a complex and diverse material family, they all share a layered lattice structure. This curious fact prompts the question of whether high-temperature superconductivity can exist in an isolated monolayer of copper oxide, and if so, whether the two-dimensional superconductivity and various related phenomena differ from those of their three-dimensional counterparts. The answers may provide insights into the role of dimensionality in high-temperature superconductivity. Here we develop a fabrication process that obtains intrinsic monolayer crystals of the high-temperature superconductor Bi2Sr2CaCu2O8+δ (Bi-2212; here, a monolayer refers to a half unit cell that contains two CuO2 planes). The highest superconducting transition temperature of the monolayer is as high as that of optimally doped bulk. The lack of dimensionality effect on the transition temperature defies expectations from the Mermin-Wagner theorem, in contrast to the much-reduced transition temperature in conventional two-dimensional superconductors such as NbSe2. The properties of monolayer Bi-2212 become extremely tunable; our survey of superconductivity, the pseudogap, charge order and the Mott state at various doping concentrations reveals that the phases are indistinguishable from those in the bulk. Monolayer Bi-2212 therefore displays all the fundamental physics of high-temperature superconductivity. Our results establish monolayer copper oxides as a platform for studying high-temperature superconductivity and other strongly correlated phenomena in two dimensions.