The £650m building, behind St Pancras station and the British Library, will be home to 1,250 scientists.
Named after the co-discoverer of DNA structure, the Crick will examine fundamental questions about how illnesses develop, in order to find new ways to treat conditions such as cancer, heart disease, stroke, infections and diseases of the brain.
The director of the institute, Sir Paul Nurse, said the Crick would "attract brilliant, bold and creative scientists from the UK and around the world".
Sir Paul, a Nobel Prize winner and former president of the Royal Society, is among the first scientists to move into the building.
He and his team are studying cell division, which is crucial to understanding how cancer and other diseases develop.
Sir Paul said the UK's decision to leave the EU meant the Crick would lose a planned £10m a year in EU funding, which would need to be replaced by government funding.
However, he said, Britain was a "great scientific nation" and the Crick would be a symbol the UK was "open for business".
The Crick is funded mainly by the Medical Research Council, Cancer Research UK and the Wellcome Trust.
The building will replace ageing and cramped laboratories at Mill Hill in north London, Lincoln's Inn Fields, and Potters Bar, Hertfordshire.
Although the main focus of the Crick will be the biological processes underlying human health and disease, it will also do "translational research", which aims to turn lab discoveries into treatments for patients.
The institute has a partnership with pharmaceutical giant GlaxoSmithKline.