Billions of functionally distinct blood cells emerge from a pool of hematopoietic stem cells in our bodies every day. This progressive differentiation process is hierarchically structured and remarkably robust. We provide an introductory review to mathematical approaches addressing the functional aspects of how lineage choice is potentially implemented on a molecular level. Emerging from studies on the mutual repression of key transcription factors, we illustrate how those simple concepts have been challenged in recent years and subsequently extended. Especially, the analysis of omics data on the single-cell level with computational tools provides descriptive insights on a yet unknown level, while their embedding into a consistent mechanistic and mathematical framework is still incomplete.