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Open the Instagram app on your smartphone. Open your profile if it isn't already open. Open your account's settings menu. Scroll down to the "Account" group. Swipe the switch next to "Private Account" on.
Switching your account from its standard status to "Private" means that anyone wishing to follow you will have to request to do so; you are the only person who can approve said request. This gives you greater control over who can access your Instagram. Changing your account to "Private" also restricts user access to your comments and likes, with the sole exception being for public posts (wherein your name will appear next to other "likes", but your account will still be protected). You won't be able to change your account's viewing status from a computer. To do this, tap the person-shaped icon in the bottom right corner of your phone's screen. You can also do this on a tablet. Do this by tapping the gear icon (iOS) or the three dots (Android) in the top right corner of your screen. This is a series of tabs dedicated to account options; you'll find the "Private Account" option at the bottom of this group. It should from grey to blue, indicating that your account is now private! If you want to disable this setting, simply swipe the switch back and tap "OK" on the pop-up window. Note that your current followers will not be affected by this switch. If you want to block some or all of them, you'll need to do so manually.