Mining is a way to validate transactions in a way that is decentralized. This is important for the integrity of a permissionless blockchain where users are anonymous. But you could easily create a permissioned blockchain, like Hyperledger, where all of the member nodes are known and trusted. This type of blockchain would not require mining.
Mining is a process of solving the complex mathematical problem in order to validate the transactions and add that block to the existing blockchain. There are companies who are working on different projects to implement blockchain without miners.