Atomically precise metal nanoclusters (MNCs) have gained tremendous research interest in recent years due to their extraordinary properties. The molecular-like properties that originate from the quantized electronic states provide novel opportunities for the construction of unique nanomaterials possessing rich molecular-like absorption, luminescence, and magnetic properties. The field of monolayer-protected metal nanoclusters, especially copper, with well-defined molecular structures and compositions, is relatively new, about two to three decades old. Nevertheless, the massive progress in the field illustrates the importance of such nanoobjects as promising materials for various applications. In this respect, nanocluster-based catalysts have become very popular, showing high efficiencies and activities for the catalytic conversion of chemical compounds. Biomedical applications of clusters are an active research field aimed at finding better fluorescent contrast agents, therapeutic pharmaceuticals for the treatment and prevention of diseases, the early diagnosis of cancers and other potent diseases, especially at early stages. A huge library of structures and the compositions of copper nanoclusters (CuNCs) with atomic precisions have already been discovered during last few decades; however, there are many concerns to be addressed and questions to be answered. Hopefully, in future, with the combined efforts of material scientists, inorganic chemists, and computational scientists, a thorough understanding of the unique molecular-like properties of metal nanoclusters will be achieved. This, on the other hand, will allow the interdisciplinary researchers to design novel catalysts, biosensors, or therapeutic agents using highly structured, atomically precise, and stable CuNCs. Thus, we hope this review will guide the reader through the field of CuNCs, while discussing the main achievements and improvements, along with challenges and drawbacks that one needs to face and overcome.