A man has died after 35 people - including children - were found in a shipping container at Tilbury Docks.
The survivors - believed to be from the Indian subcontinent - are said to be recovering "fairly quickly in most cases" at nearby hospitals.
They were discovered after a freighter arrived from Zeebrugge, Belgium at about 06:00 BST and was being unloaded.
Essex Police have launched a homicide investigation and officers are being assisted by their Belgian counterparts.
Supt Trevor Roe said staff at the docks were alerted to the container by "screaming and banging" from inside.
He said about 50 other containers on the freighter, called the Norstream, had been searched and no other people were discovered inside.
Speaking at a press conference, Mr Roe told journalists the survivors were being held under immigration powers and would be taken to an immigration reception centre near Tilbury.
He said they would eventually be interviewed through interpreters.
Belgian police say they believe the lorry which delivered the container in Zeebrugge has been identified through CCTV footage. They do not, however, have information yet about where it originated from.
Mr Roe said the police investigation would look into "the gangs or whoever may be involved in this conspiracy to bring these people in this way over to this country".
It is not known where the container, one of 64 aboard the P&O vessel, originated. Mr Roe also said he did not know where the survivors had been going.
At the scene
Ben Ando, BBC News correspondent
At Basildon hospital police vans were parked in between rows of ambulances in a clash of high-visibility stripes and chevrons.
At the dropping-off zone in front of the entrance, patients came and went, glancing up at the news helicopter circling overhead.
Most were aware of the high-security patients being treated in a cordoned-off section of the accident and emergency unit; the atmosphere became more relaxed when it became clear that their conditions were primarily dehydration and hypothermia, and nothing more serious or contagious.
Immigration and security minister James Brokenshire said the incident was "a reminder of the often devastating human consequences of illegal migration".
He said: "We know that criminal gangs are involved in what amounts to a brutal trade in human lives. We also know that illegal migration is a Europe-wide issue.
"That is why we work closely and collaboratively with law enforcement and port authorities, in neighbouring countries, to target criminal networks and ensure that the organised gangs behind trafficking and people smuggling can't operate with impunity."
The East of England Ambulance service, which was called to the docks at 06:37 BST, sent seven ambulances, two rapid response cars, two doctors and a hazardous area response team to the scene.
Assistant Chief Officer Daniel Gore from the ambulance service said none of the people being treated in hospital was thought to have life-threatening conditions.
Part of Basildon Hospital, where 18 people from the container were taken for treatment, was taped off earlier.
On its website, the hospital said its accident and emergency department was "responding to a major incident".
Seven patients were taken to Southend Hospital while nine were taken to the Royal London Hospital, Whitechapel.
South Basildon and East Thurrock MP Stephen Metcalfe described the incident as "tragic".
Mr Metcalfe told the BBC: "The fact that so many people appear to have travelled so far and are so desperate to get into the UK - either on their own or being trafficked is really very sad."
The Conservative MP said it was important "to get to the root causes of what is motivating people to go to such extreme lengths to travel from other parts of the world to get into the UK" as well as tackle people trafficking.
'Noise from container'
The container was loaded on to the P&O freighter at about 21:30 BST on Friday at Zeebrugge.
A P&O spokeswoman said the Norstream, which had been scheduled to leave Zeebrugge at 22:00 BST on Friday, was also carrying 72 trailers and five lorries and their drivers.
Public Health England said it was not currently involved and had not been notified of there being any Ebola risk.
Essex Ambulance said decontamination units had been set up at the docks as a precaution, not because of any specific concerns it had.
A spokeswoman for the port declined to comment on the incident as it "was a matter for the police and Border Force".
The ship was sailing on a new service linking Tilbury and Zeebrugge which has only been operational since earlier this month, according to parent company Forth Ports.
According to the UK Border Force, the number of "clandestine illegal entry attempts" by people to enter the country via ports in Belgium and France increased last year to 18,000 from about 11,000 in the previous 12 months.
The Freight Transport Association said it was "quite unusual" for stowaways to be found in containers, with most cases involving people attempting to enter the UK on lorries.
In June 2000, the bodies of 58 Chinese people were found in the back of a lorry smuggling them into Dover on a ferry from Zeebrugge. They had died of suffocation.