In a combat encounter, if you fail to kill the monster, can you keep going until you defeat it or do you have to stay there and wait until the next turn? Also, do you have to do a horror check every time? I know in Arkham you only had to do it once.

A combat encounter in EH (if possible, forget about Arkham, since the games, even if sharing a similar settings, have rather different rules, especially regarding gates and combat) implies one investigator attempting to defeat each of the monsters in his space. A combat encounter consists of two checks: a single Will check, and a single Strength check.

When you attempt the Strenght check, the monster loses a number of Health points equal to the number of successes rolled (and you could be damaged in the process if the number of successes rolled is lower than the monster's damage). If this kills the monster, all the better. Otherwise the monster stays where it is. If there is another investigator in the same space, this second investigator must attempt another Combat Encounter and so on.

Health damage inflicted to the monster is carried over to the following round, where, in case investigators are still in the same space as the monster (or new investigators arrive) everything is repeated again exactly as in the first round of combat: Will check, Strength check.

There's a good example of combat on pag 15 of the rulebook, in the lower section of the page.

Hope this clarifies your doubts

...and the really cool thing is you can both injure the monster and you only take damage yourself if the delta between the damage you inflict and that of the monster is a positive number. Thus, if you inflict "2" points of damage and the monster would inflict "3" the Investigator takes "1" point of Health damage.

Julia, please correct any of my ramblings...

If you fail to kill the monster on the first go, then you are done encountering the monster for that turn. Total up any losses and you are good to go. You do have to make Will each turn/encounter as well.

The "will test per encounter" issue is an Arkham Horror red herring/hangover that I wish we could just get off the table. All one has to do is execute a Combat Encounter each turn that one is in the same location as a Monster, each CE is a discrete event. With EH pretty much all you have to do is play the cards as they are written and execute the rules in a like manner. It's one of the strengths of the game (despite the occasional misstep along the way)

Thank you very much for the help. I think i've got it now. ^_^x