In 1607, Hudson set sail on the Hopewell to find a northeast passage to Asia through the Arctic Ocean via the North Pole. The voyage was paid for by the Moscovy Company, one of a small number of corporations given Royal Charters. In June he reached the eastern shore of Greenland and started northward, mapping as they went. On the 20th they started out for Svalbard , eventually reaching an island on the northern end of the group on the 17th of July. At this point the ship was only 577 nautical miles from the pole, but it was clear there is no way to go further due to the ice and he decided to return to England on the 31st. On the return voyage Hudson discovered what is now known as Jan Mayen Island before reaching home in September.
In 1608 he tried again, this time sailing farther to the east along the northern coast of Norway. Once again all northern routes were blocked by ice and he ended up reaching Novaya Zemlya before turning back. This point had been reached by several crews in the past and was considered the end of the line, which convinced the Moscovy Company that there was no point funding further voyages.
At this point Hudson wanted to continue his explorations and turned to the Dutch East India Company for funding. They were particularly interested in shorter routes to the east, and commissioned a new ship for his use, the Halve Maen (Half Moon). The ship headed north in May 1609, but was forced to turn back before reaching Novaya Zemlya.
Instead the expedition headed west and eventually reached the Grand Banks, off Newfoundland, in early July. They spent the next four months exploring the east coast of North America, including Manhattan, Maine, and Cape Cod–the first Europeans to describe these locations (although Giovanni da Verrazano explored the same coast in 1524)–and sailing a distance up the Hudson River, which bears his name. The Dutch would later claim the area and set up a colony as New Amsterdam.
Upon returning to Europe in November they made port at Dartmouth, where Hudson was arrested for sailing under another country's flag. He was soon released.
In 1610 Hudson managed to get backing for yet another voyage under the English flag; this time the funding came from the Virginia Company and the British East India Company. At the helm of his new ship, the Discovery, he stayed to the north (some claim he deliberately went too far south with the Dutch), reaching Iceland on May 11, the southern end of Greenland on June 4, and then managing to turn around the southern tip of Greenland and continue west.
Excitement was high that a ship had finally found the Northwest Passage, and on June 25th they reached Hudson Strait at the northern tip of Labrador. Following the southern coast of the strait the ship sailed into Hudson Bay and spent the next months continuing to map and explore the eastern shores. In November the ship became trapped in the ice in James Bay, and the crew moved ashore for the winter.
When the ice cleared in the spring of 1611, Hudson wanted to continue exploring, but the crew wanted to return home. Eventually matters came to a head and the crew mutinied, setting Hudson, his son, and several other crew adrift in a small boat. They were never seen again, although some claim that he successfully made his way as far south as the Ottawa River.
Apparently the reports by Hudson of his voyage for the Dutch have been lost, but an account was given by Johannes de Laet in his work Nieuwe Wereldt ofte beschrijvinghe van West-Indien (first edition 1625). The same situation applies to the voyage of Adriaen Block.