In this paper, we review research on homeopathy from four perspectives, focusing on reviews and some landmark studies. These perspectives are laboratory studies, clinical trials, observational studies, and theoretical work. In laboratory models, numerous effects and anomalies have been reported. However, no single model has been sufficiently widely replicated. Instead, researchers have focused on ever-new models and experiments, leaving the picture of scattered anomalies without coherence. Basic research, trying to elucidate a purported difference between homeopathic remedies and control solutions has also produced some encouraging results, but again, series of independent replications are missing. While there are nearly 200 reports on clinical trials, few series have been conducted for single conditions. Some of these series document clinically useful effects and differences against placebo and some series do not. Observational research into uncontrolled homeopathic practice documents consistently strong therapeutic effects and sustained satisfaction in patients. We suggest that this scattered picture has to do with the fourth line of research: lack of a good theory. Some of the extant theoretical models are reviewed, including placebo, water structure, silica contamination, energy models, and entanglement models. It emerges that local models, suggesting some change in structure in the solvent, are far from convincing. The nonlocal models proposed would predict that it is impossible to nail down homeopathic effects with direct experimental testing and this places homeopathy in a scientific dilemma. We close with some suggestions for potentially fruitful research.