I'm new to Azure deployments. We have installed Sitecore 8.2 webapp and deployed and everything works fine except that our custom indices have not been created. Pages that use indices crash. Rebuilding indices through Sitecore completes right away and the folder under App_Data does not get created. We're using Lucene. Are we missing something?
The default installation of Sitecore on Azure PaaS is using Azure Search. Lucene indexing is not supported as it is a file based solution. In PaaS you either have to use Azure Search or Solr or Coveo.
Make sure to update your custom index config file before publishing and rebuilding the index.
If you are using Azure Search, please read the features and limitations: https://doc.sitecore.net/sitecore_experience_platform/setting_up_and_maintaining/search_and_indexing/sitecore_azure_search_overview