See also: Creating a custom Azure Logic Apps connector on my blog.
History: BizTalk → SharePoint workflow manager → Logic Apps.
Connectors are⇒ Azure API Apps are⇒ Web API apps.
Actions - “Regular old Web API
Creating a custom connector
Basic steps (taken from support forum)
Install the Azure SDK
Create a API
App in Visual Studio (New Project → Web → API
Implement a controller that contains your method
Enable Swagger: Goto App_Start/SwaggerConfig and Remove the comment before EnableSwaggerUI (About line 176).
Publish to the cloud (either create an API
App on portal.azure.com and select it in vs or create a new one on publish).
You can then select your API
App from a Logic App.