Non-custodial

Basics

Sometimes styled as self-custodial.
One of the terms often used in the crypto space when it comes to wallets and smart contract solutions. Custodial is the opposite of non-custodial, like permissioned and permissionless, centralized and decentralized, and private blockchains and public blockchains. It is all about who has the full ownership of the underlying assets, and in the case of the former examples, custodial, permissioned, centralized and private blockchains have a third party that can access your assets. With the non-custodial, permissionless, decentralized and public blockchains, these assets should in theory only be accessible by the user itself.
From this blog (30-8-2020):

"Non-custodial wallets allow their users to fully manage and own their private keys and data. During the setup process the wallet will let you download a private key file or a mnemonic phrase.

Pros:

Full Control

Users have full access and control over their funds, and no third party handles their assets.

Safety

Since all of the details associated with users’ wallets and funds rely on themselves, there is a much lower risk of a data breach.

“Instant” Withdrawals

Transactions don’t require third party approval. This streamlines the entire process and makes the withdrawal process faster.

Full forks/airdrops support

Users can get access to any fork or airdropped crypto-asset they are eligible to. Even if the wallet that they are currently using doesn’t support one of the assets, they can quickly get access to them by importing their address to the wallet that supports them (since they have their private key or mnemonic phrase).

Participation in consensus

Non-custodial wallet users can participate in PoS consensus by staking/voting/delegating their assets. They are free to choose which validator they want to support.

Cons:

Potential loss of access

Losing private key or mnemonic seed can result in losing access to the wallet, with no possibility of backup and recovery as there is no centralized agent with access to this data. Key and mnemonic phrase management is entirely up to the user.

Network fees

All transactions are reflected on the chain in real-time, and users will have to pay commission fees for each transaction."