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If I have understood it right Mechanized forces have a reduced effectivnes fighting in urban areas? Are there other effects depending on terrain? Are infantry fighting better in forests or marsh than armor and mechanized?

simpley check the properties of a unit and the properties of terrain types.
There you see the attack/defend values and for terrain the defense bonus for each unit category.

But don't forget that terrain with good defense also reduces supply most of the time. A forrest next to a supply 10 city without a road has only 8 supply instead of 9 --> minus 2 per hex. For mountains that value is even lower! I think minus 3 per hex. So fighting somewhere in rural areas might give you a bonus but keep in mind to have an eye on your supply. Getting destroyed at supply <5 is not what you want.

Thanks - will try to place the unit in respective terrain and take a closer look. Yeah HQ is necessary to make sure those units are not cut down because of the supply situation. I could not find any god explanation in the rule book. Is there any good tables on some forums or some where else for combat with all modifiers and units represented? I have read the one in the rule book, but that does not include terrain modifiers for example.

There is just the formula that is more or less self explaining. Since this formula is a bit more complex it is a bit hard to make any examples. But to know where to get a good bonus is enough.

In SC2 it was best to place a tank in a city. In SC3 armies get a better bonus. Took me some games to get it