Version 2.0 will definitely load your tournaments from 1.3.5 (and earlier). It will automatically convert as much as it can to the new format. There are a number of areas you'll want to possibly check/update after it does the automatic conversion:
1 - Players: Obviously there was no database in previous versions, so none of the players will be database players. However, upon loading the tournament, the TD will attempt to match players in the tournament with players in your database. The good thing here is that if you go through the steps of converting one 1.3.x tournament, and adding those players to the database, then when you convert other tournaments, the TD should be able to automatically match players with players in your database. Assuming of course you use the same names for players across different tournaments.
2 - Prizes: Prizes had a name and a value in 1.3.x, so they were never actually "awarded" to players. The TD will attempt to determine which prizes go with which ranks, and convert them to ranked prizes so they are automatically "awarded". You should definitely check to make sure it made the right assumptions after converting your tournament.
3 - Layouts: Layouts will import, but Tim is right, especially here. If you actually want to use a layout you've imported from 1.3.x, you'll probably want to do some work on it, as the layouts are much more extensive in 2.0.