Routes affected include lines into London Waterloo via Surbiton, South Westsuburban routes including Shepperton to Fulwell and Virginia Water to Weybridge, lines into London Victoria from East Croydon, and metro routes in South East London, including those running via Bexleyheath and Blackheath.
Disruption has been caused by a lightning strike at Surbiton, impacting routes into Waterloo. There was flooding on Wandsworth Common, affecting routes into Victoria, as well as flooding at Bexleyheath, Catford and on the Redhill-Tonbridge line.
Staff have been out through the night trying to contain the chaos, Network Rail said, and are still working to keep the network running.
A Network Rail spokesperson said: "It's a very fast-moving picture this morning, with flash floods still arising and other falling in a variety of different routes into London.
"In addition, we are dealing with the aftermath of lightning strikes at Surbiton, which is on the main line into Waterloo, and at Blackheath.
"Our advice to passengers in the South East this morning is to please check before you travel, either with your operator's website or National Rail Enquiries.