Core-collapse supernovae (CCSNe) mark the deaths of stars more massive than about eight times the mass of the Sun and are intrinsically the most common kind of catastrophic cosmic explosions. They can teach us about many important physical processes, such as nucleosynthesis and stellar evolution, and thus they have been studied extensively for decades. However, many crucial questions remain unanswered, including the most basic ones regarding which kinds of massive stars achieve which kind of explosions, and how. Observationally, this is related to the open puzzles of whether CCSNe can be divided into distinct types or whether they are drawn from a population with a continuous set of properties, and what progenitor characteristics drive the diversity of observed explosions. Recent developments in wide-field surveys and rapid-response follow-up facilities are helping us answer these questions by providing new tools, such as: (1) large statistical samples that enable population studies of the most common SNe and reveal rare (but extremely informative) events that question our standard understanding of the explosion physics involved; and (2) observations of early SNe emission taken shortly after explosion, which carry signatures of the progenitor structure and mass-loss history. Future facilities will increase our observational capabilities and allow us to answer many open questions related to these extremely energetic phenomena of the Universe.