Welcome to iMore! Are these iMessages or SMS? Or both?
Has it worked before? Did this just start? Have you tried a hard reset?
Hold the top or side power/sleep button and the home button at the same time until the Apple logo appears, then release.
Have you tried going to Settings > Messages > Toggle iMessage off, then back on?
Have you tried signing out of iMessage, then sign back in?
If you've done all these things and the problem persists, try Reset All Settings.
Reset All Settings
Go to Settings >General >Reset >Reset All Settings. This resets to default everything in Settings, including notifications, alerts, brightness, and clock settings like wake up alarms.