The Geminga pulsar, one of the brighest gamma-ray sources, is a promising candidate for emission of very-high-energy (VHE > 100 GeV) pulsed gamma rays. Also, detection of a large nebula has been claimed by water Cherenkov instruments. We performed deep observations of Geminga with the MAGIC telescopes, yielding 63 h of good-quality data, and searched for emission from the pulsar and pulsar wind nebula. We did not find any significant detection, and derived 95% confidence level upper limits. The resulting upper limits of 5.3 × 10-13 TeV cm-2 s-1 for the Geminga pulsar and 3.5 × 10-12 TeV cm-2 s-1 for the surrounding nebula at 50 GeV are the mostconstraining ones obtained so far at VHE. To complement the VHE observations, we also analyzed ...