Head and neck squamous cell carcinoma (HNSCC) represents a highly prevalent and deadly malignancy worldwide. The prognosis for locoregionally advanced HNSCC has not appreciably improved over the past 30 years despite advances in surgical, radiation, and targeted therapies and less than 20% of HNSCC patients respond to recently approved immune checkpoint inhibitors. The Hippo signaling pathway, originally discovered as a mechanism regulating tissue growth and organ size, transduces intracellular and extracellular signals to regulate the transcriptional co-activators YAP and TAZ. Alterations in the Hippo pathway resulting in persistent YAP and TAZ activation have emerged as major oncogenic drivers. Our analysis of the human HNSCC oncogenome revealed multiple genomic alterations impairing Hippo signaling and activating YAP and TAZ, which in turn contribute to HNSCC development. This includes mutations and deletions of the FAT1 gene (29%) and amplification of the WWTR1 (encoding TAZ, 14%) and YAP1 genes (8%), together representing one of the most genetically altered signaling mechanisms in this malignancy. Here, we discuss key elements of the mammalian Hippo pathway, detail mechanisms by which perturbations in Hippo signaling promote HNSCC initiation and progression and outline emerging strategies to target Hippo signaling vulnerabilities as part of novel multimodal precision therapies for HNSCC.