All of the previous schemes have relied on a single private key to control the wallet. This presents an all-or-nothing risk for loss of funds from both theft and negligence. To counter this, a wallet can have several private keys attached of which all or a subset need to sign any transactions.
This is often called multi-signature, or multisig for short, but is also sometimes referred to as a vault. A multi-key setup is described as n-of-n to indicate how many keys are needed to sign a transaction out of the issued number. For example, a 2-of-3 setup requires two of the three private keys to sign a transaction for it to be valid.
In the case of a personal wallet, one individual will control all the keys but hold them on different devices for increased security. See the savings wallet reference design for a UX exploration of this use case.
In the case of a shared wallet, different people will control the keys. The number of keys and required co-signers will depend on the use case. With spouses sharing a joint account, a simple 1-of-2 multi-key setup might suffice, meaning there are two keys but only one is required to sign for a transaction to be valid. At the other end of the spectrum, a company might require a more complex 3-of-5 setup, requiring three of the five co-signers to approve any transaction.
It is important to remember that the usage of multiple keys is necessary only for signing outgoing transactions, but not for receiving funds. This is a common misunderstanding.
Multi-key schemes can raise the security, since anyone needs access to more than one key to move any funds. But clearly, it also increases complexity and the requirements on the user to keep even more keys securely stored and/or backed up.
Tip
The keys used by a lightning node cannot be controlled by a multi-key setup, as they need to be continuously available to the node.
A software wallet application or coordination software initiates a multi-sig wallet, choosing the number of total keys, and the number required to sign transactions. You then add private keys from other wallets generated elsewhere to the multisig after which the software wallet can complete the creation process. For any future transaction from the multi-sig wallet, the required amount of co-signers need to sign (using Partially Signed Bitcoin Transactions - PSBT from BIP174) before any transaction is valid.
Pros
Significantly increases security against theft
Can allow several people to access and control a shared wallet
Can tailor requirements for multiple co-signing and access situations
Cons
Has significant complexity and op-sec burden for multiple private keys, each of which needs a good backup scheme