With the rapid development of mobile devices, smartphones have gradually become an indispensable part of people's lives. Meanwhile, biometric authentication has been corroborated to be an effective method for establishing a person's identity with high confidence. Hence, recently, biometric technologies for smartphones have also become increasingly sophisticated and popular. But it is noteworthy that the application potential of palmprints for smartphones is seriously underestimated. Studies in the past two decades have shown that palmprints have outstanding merits in uniqueness and permanence, and have high user acceptance. However, currently, studies specializing in palmprint verification for smartphones are still quite sporadic, especially when compared to face-or fingerprint-oriented ones. In this paper, aiming to fill the aforementioned research gap, we conducted a thorough study of palmprint verification on smartphones and our contributions are twofold. First, to facilitate the study of palmprint verification on smartphones, we established an annotated palmprint dataset named MPD, which was collected by multi-brand smartphones in two separate sessions with various backgrounds and illumination conditions. As the largest dataset in this field, MPD contains 16,000 palm images collected from 200 subjects. Second, we built a DCNN-based palmprint verification system named DeepMPV+ for smartphones. In DeepMPV+, two key steps, ROI extraction and ROI matching, are both formulated as learning problems and then solved naturally by modern DCNN models. The efficiency and efficacy of DeepMPV+ have been corroborated by extensive experiments. To make our results fully reproducible, the labeled dataset and the relevant source codes have been made publicly available at https://cslinzhang.github.io/MobilePalmPrint/.