How can I see which campaigns a lead is currently in?

Not precisely. If you click on a particular lead, you can view its activity, which should show you which campaigns they are members of.

Thats a pitty. Leads can have several pages in the activity log. Will take ages. 
Thanks Josh

Hi  Ingibjorg, what you want to do is use the Filter in the upper left hand corner.

From the drop down select "Program Status" and it will show you only the events with an activity type of "Change Program Status" so you'll be able to see the campaigns the lead moved through and each time they changed status in those campaigns.

You can also create your own custom filter as well. Just remember to change the filter setting back to "None" when you're done. I've scared myself in the past by forgetting I turned it on, gone to check a lead and went "Oh no! Everything's gone! Wait - no, it's just filtered. Filter -> None. There it all is!"

Try using a Leads by Campaign report. 

Set "Created at" to "All Time" in the setup tab. 

In the smart list, choose filter "Email Address" and enter the email of the lead you are interested in.