Is there any way in camunda task-list to get list of the processes I have started in order to see the progress

Historical task rest api will give you both completed and active task instances. It won’t return the task instances if the execution token has not reached that activity.

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If you are looking for only active task instances then you can query directly on task rest api instead of history api. Historical data will have huge records, so table scan and query execution might take more time compared to runtime task instance.

Only for active(runtime) task instances:
https://docs.camunda.org/manual/7.11/reference/rest/task/post-query/

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If you want to notify the user for the tasks assigned to him, you can configure task Listener with the event type=assignment, it will notify the user that a action is required by you in order to proceed the task.

https://docs.camunda.org/manual/7.11/user-guide/process-engine/delegation-code/#task-listener

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