Deribit Derivatives Exchange operates a maker-taker fee model. Futures orders which provide liquidity receive a rebate of 0.02%. Orders that take liquidity are charged a small fee of only 0.05%. The fee is calculated as a percentage of the underlying asset of the contract. Perpetual contracts orders providing liquidity receive a rebate of 0.025%, order that take liquidity are charged a fee of 0.075%.

BTC Perpetual Contracts
Maker Rebate: 0.025%
Taker Fee: 0.075%

BTC Futures
Maker Rebate: 0.02%
Taker Fee: 0.05%

ETH Perpetual + Futures 
Maker Rebate: 0.00%
Taker Fee: 0.00%

BTC and ETH Options
0.04% of underlying or 0.0004 BTC/option contract
0.04% of underlying or 0.0004 ETH/option contract

Fees can never be higher than 12.5% of the price of the option. For example if an option is traded at 0.0001 BTC, the taker fee will be 0.0000125 BTC (instead of 0.0004 BTC), thus 12.5% of 0.0001 BTC.

Delivery (settlement at expiration)
For deliveries (expiration) half the fees of taker orders are charged.  Eg: 0.025% for Futures, and 0.02% for Options, where for options the fee can never be more than 12.5% of the value of the option.

Fees for Liquidation trades.
Liquidations are charged a higher fee than normal. Fees are raised with 0.30% of underlying contract size. The surplus fees of liquidations will automatically fund the insurance fund.
Futures liquidations fees:
0.35% fee,  0.30% goes to the insurance fund.
Perpetual contracts liquidations fees:
0.375% fee, 0.30% goes to the insurance fund

Options liquidations trades: 0.19% of underlying or 0.0019BTC/options contract, 0.15% of underlying or 0.0015BTC per contract goes to the insurance fund.
So the higher fees of liquidation trades are the income for the insurance fund.

Deposits and Withdrawals
Bitcoin deposits are credited after 1 confirmation on the network. We don't charge fees for deposits. Withdrawals are processed instantly if the balance in our hot wallet permits so. We keep only a small percentage of coins in hot storage, so there is a chance that your withdrawal cannot be processed immediately. If needed, once a day we replenish the balance of the hot wallet from cold storage.

Withdrawals: Withdrawal fee depends on the current state of the Bitcoin network
Deposits: Free

Makera “maker” order is an order that adds liquidity to the order book. A maker order is an order that does not execute immediately but instead goes into the order book.
Takera “taker” order is an order that removes liquidity from the order book. A taker order is an order that will execute immediately against other orders upon submission to the exchange.