I am arriving in Madrid in late afternoon on a Saturday. Somwhow need to get to Sevilla on Sunday. I thought I would stay in Madrid for Sat night when I get into town.

Are there trains or buses that run on Sundays from Madrid to SEvilla? Does anyone have any details on transportation between the two cities?

Chi-girl, there are almost 20 trains from Madrid to Sevilla on a regular sunday. I would advise you to check in www.renfe.es the schedule that fits you better, and buy your ticket in advance. There are also plenty of buses, you can find the schedule in Estación Sur's website (sorry, don't know which is it, but you can look for it in google).

Have a nice trip

I often had very good luck walking into the RENFE and buying a ticket for the next train.

The one and only one time I tried to book in advance, I was unable to get a train on a Friday that was a decent hour to travel. It was necessary for us to depart at 5:00 a.m. Otherwise, all other departures were sold out.

I recommend that since you arrive on Saturday, you take the metro to RENFE and secure your ticket a day early. If you go to a travel agent on the street, you will pay a surcharge. The metro ride should be cheaper.