The ESP8266 WiFi Module is a self contained SOC with integrated TCP/IP protocol stack that can give any microcontroller access to your WiFi network. 

Schimera has developed software for the ESP8266 WiFi Module which is free and available to download now.

ESPGadgetWebDAV

• Currently supports basic PROPFIND, MOVE, DELETE, PUT and GET requests

• Uses SPIFFS filesystem, formats filesystem on initial run. Doesn't support folders, only files in the root folder

• Tested on ESP8266-01, with 1MB flash