Beaten by horses

[Bramborough]

Just a note: OP is referring to R2, not Attila...let's not forget this is still a combined forum for both games.
@BarbaricLord, I feel your pain. Horse Archers are the one R2 unit with which I truly have a hate/hate relationship. Don't like playing with them; don't like playing against them.

As others have noted, long-range missile infantry are a good answer. Most (not all) HA's are lightly armored, so they're quite vulnerable to return fire. They typically have 125 range, whereas most slingers and archers have 150 (or at least 125 themselves). So it becomes an exercise in arranging your army formation in such a way that the enemy HA's can't get in range without getting thrashed themselves.

I would think that in most cases, by the time you get to a point in the campaign where the AI is fielding the higher-end Armored Horse Archers, then you yourself will probably be rocking well-armored infantry yourself, and the HA's fire won't be as devastating to them.

Hooahguy's tactic of chasing them away with decent melee cavalry can work as well; obviously it does the job for him. I'm just not very good at managing that approach; most of the time I either manage to get my cavalry mauled...or some other unit elsewhere that I didn't pay enough attention to because I was micro-ing the cavalry.

[Slaists]

Well, if the OP is talking about R2 horse archers, cornering those with melee cav is a suicide since all the R2 horse archers get Parthian shot (360 degree shooting while running around).

In R2, massed foot archers work very well against HA's, even the armored ones.