How much does the bridge charge for a transaction?

The bridge itself doesn’t charge any fee.

When assets are inside the Aurora network, the fee is 0.03 Gwei. However, if you use Aurora Pass mobile wallet, you get 50 free transactions per month before you have to purchase transaction pack. To start using Aurora Pass wallet, visit https://auroracloud.dev/pass and download on your preferred device.

When assets are inside the NEAR network, the fees of NEAR validators are negligible (paid in NEAR, costs cents).

When assets are inside the Ethereum network, fees may be hefty (paid in ETH, the cost may go up to USD300 per transaction).

However, you can use https://app.near-intents.org to bridge the tokens from Near/Aurora to other supported chains with a minimal fee by depositing and withdrawing to the desired chain.