Custody is a safeguarding service offered by the financial institution towards customers' security. The institution providing custody is often termed as 'custodian'. They are responsible for collecting interests, dividends, and profits from the security sales while redistributing the funds based on customers' instructions.
Assets nowadays can be held in both physical and digital form by custodians. The custodial businesses are usually large, comprising respected entities, like banks, exchanges, etc. Custodians preserve the assets for a large number of customers, which even sums up to billions of dollars in many cases.
In the blockchain world, the custody of cryptocurrencies is given to crypto exchanges. They solely hold the private keys along with crypto assets for their users. The custodians of such crypto assets are often known as 'crypto custody providers.' With the majority of the user portfolio held in 'cold storage,' this allows custodians to safeguard its customer's crypto assets in case of any mishappening.