You could use GET /rest/api/2/search
and use the jql
parameter to specify the search you want to do.
This is a link to the relevant section of the API documentation: https://docs.atlassian.com/jira/REST/cloud/#api/2/search-search
So, assuming you want to get JSON for issues assigned to current user you can visit:
https://your-jira-name.atlassian.net/rest/api/2/search?jql=assignee%20%3D%20currentUser()
Note: this will be limited to first 50 issues by default. Read the 'Query Parameters' section in the API docs for more info.
Sam
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