Domain experts are desperately looking to solve decision-making problems by designing and training Machine Learning algorithms that can perform classification with the highest possible accuracy. No matter how hard they try, classifiers will always be prone to misclassifications due to a variety of reasons that make the decision boundary unclear. This complicates the integration of classifiers into critical systems, where misclassifications could directly impact people, infrastructures, or the environment. The paper proposes to consider a classifier as a structural part of the system instead of an individual component to be tested in isolation and included in the system afterward. This allows for omitting those predictions that are suspected to be misclassifications, triggering system-level mitigation strategies. The resulting fail-controlled classifiers (FCCs) are software components that can correctly classify, misclassify, or omit outputs: ideally, they would omit all and only outputs that correspond to misclassifications. After presenting the theoretical foundations of FCCs, the paper proposes metrics to quantify their performance, 5 software architectures for FCCs, and an experimental analysis involving tabular data and image classifiers. Overall, this paper advocates the need for a system and software design in which ML classifiers are not separate components, but should rather be considered building blocks that interact with other components for improved performance.