I'm assuming you've recently upgraded your phone to Ice Cream Sandwich and this reply is relevant to that.
Reading the forum you'll find there seems to be two dominating problems with the wifi capabilities of ICS. Firstly, you could have a connection problem - the phone tries to connect to a wifi network, but constantly drops it before it can connect. Secondly, the phone connects to your wifi network, but you can't actually use the internet. It's slow, sporadic, sometimes drops your wifi connection and gives lots of errors.
The second problem is what I've been having trouble with and it sounds to me like that's your problem too. You can have a look at the topic in which we've discussed it - http://talk.sonymobile.com/thread/35992?tstart=30
All we can hope for is that Sony either gets an update that fixes it out very soon, or gives us the option to downgrade to Gingerbread. I've got a feeling the update will take a while, so I'm personally hoping for the option to downgrade.
