In ASP.NET, you can easily use SignalR and provide a push messaging service to your clients who are using a web app version, not the mobile or desktop version. The SignalR would provide a very easy way to enable push messaging, such as the chat notifications or messages whenever there is a new message on the server hub. SignalR uses hub and clients, to notify the people of anything new in the system, and thus it
pushes that update to the clients.
For SignalR please read the following,
SignalR | The ASP.NET Site[
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There are a few other ways to implement this in the mobile or client devices, they require some sort of service such as Azure Notification Hub, to be running and that sends the notifications to your client devices. Please read my following answers for that,
Real Time Web functionlity[
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How to implement push notifications[
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