Can Appian launch a process based on an SMS message from an external application

You would need to send the SMS in a way Appian can pick-up.. such as Email, Web Service, FTP File Drop, Database, etc..

If you can drop the SMS messages in some storage medium (Database, File, etc..) you can design a process to periodically poll the storage to discover new SMS messages.

Yes, Appian can launch a process based on an SMS message from an external application. Here's a general idea of how you can achieve this:

1. Create an HTTP endpoint in Appian that accepts incoming SMS messages.
2. Integrate the external SMS application with Appian using the HTTP endpoint.
3. Parse the SMS message in Appian to extract the relevant information.
4. Use the extracted information to trigger a process in Appian, such as creating a new record, updating an existing record, or starting a new workflow.

You may need to write custom code in Appian or use a plugin to handle the SMS integration, but the exact implementation will depend on the specifics of your use case and the SMS application you're using.

Read More: SMSala

The original request is 9 years old. Lot has changed then since. 
https://docs.appian.com/suite/help/22.4/ccs-21.4.1.0/ccs-site-features.html

Twilio Utilities

Appian Connected Claims offers connection capabilities with Twilio Utilities. If you have Twilio Utilities services set up, you can integrate them into your Connected Claims solution to add voice, chat, and SMS messaging capabilities to the solution using the Twilio service. Provide your customers with multiple ways to communicate with you. Use your customer data to track all of your customers interactions. This integration requires you to have your own Twilio account.