Over the past two decades, biometric recognition has exploded into a plethora of different applications around the globe. This proliferation can be attributed to the high levels of authentication accuracy and user convenience that biometric recognition systems afford end-users. However, in-spite of the success of biometric recognition systems, there are a number of outstanding problems and concerns pertaining to the various sub-modules of biometric recognition systems that create an element of mistrust in their use - both by the scientific community and also the public at large. Some of these problems include: i) questions related to system recognition performance, ii) security (spoof attacks, adversarial attacks, template reconstruction attacks and demographic information leakage), iii) uncertainty over the bias and fairness of the systems to all users, iv) explainability of the seemingly black-box decisions made by most recognition systems, and v) concerns over data centralization and user privacy. In this paper, we provide an overview of each of the aforementioned open-ended challenges. We survey work that has been conducted to address each of these concerns and highlight the issues requiring further attention. Finally, we provide insights into how the biometric community can address core biometric recognition systems design issues to better instill trust, fairness, and security for all.