When you block a phone number or contact, they can still leave a voicemail, but you won't get a notification. Messages that are sent or received won't be delivered. Also, the contact won't get a notification that the call or message was blocked.

iPhone

Source: https://support.apple.com/en-au/HT201229

If you're in the Phone app under Recents,

• tap the info icon next to the phone number or contact that you want to block.
• Scroll to the bottom of your screen, then tap Block this Caller.

Messages

If you're in Messages, open the conversation,

• tap the contact at the top of the conversation
• then the info icon
• Tap the name or phone number, scroll to the bottom of the Info screen
• tap Block this Caller.

Manage Your Blocked Phone Numbers And Contacts

To see the phone numbers and contacts that you've blocked from Phone, FaceTime, or Messages:

Phone

Go to Settings > Phone > Call Blocking & Identification.

Messages

Go to Settings > Messages > Blocked.

Android

Source: https://www.androidauthority.com/how-to-block-a-phone-number-246484/

Google Pixel

• Open the phone app
• Tap “More” (the three dots stacked vertically located on the top-right corner).
• Select Settings > Call Blocking or Blocked numbers

OR

• Open Call history or Recent Calls
• Tap the number you want to block
• Select block (you can also report call as spam)

Samsung

• Open the Phone app.
• Select which number you want to block and hit “More” (the three dots stacked vertically located on the top-right corner).
• Select “Add to Auto-Reject List.”
• To remove or make more edits, go to Settings > Call Settings > All Calls > Auto Reject.

LG Phones 

• Open the Phone app.
• Tap the three-dot icon (top-right corner).
• Select “Call Settings.”
• Select “Reject Calls.”
• Tap the “+” button and add the numbers you want to block.

HTC Phone 

• Open the Phone app.
• Press and hold the phone number.
• Select “Block Contact.”
• Select “OK.”
• You can remove them from the blocked list on the People app.

Huawei or Honor Phones 

• Open the Dialer app.
• Tap and hold a number you want to block.
• Tap “Block contact”