You have an Espionage Screen and an "Espionage Slider" (red circle under your Science and Culture sliders) that allows you to spend $$ on Espionage. Further, a number of buildings can be built in your cities that provide Espionage "points." These "points" compete between Civilizations, and if you can accumulate more against your opponant than he has against you, you'll start triggering the aforementioned "Passive Missions." (Visible in the Espionage Screen.) The more of a lead you gain in these "points," the more Passive Missions will open up.
One of the lower-level Passive Missions is that you can see what your opponant is researching (or v.v., depending on who's ahead), and further on you can see what individual rival cities are building, and later you can even double-click on and investigate rival cities.
Further, the more of a point advantage you have, the cheaper your "active" Spy Missions will become. Plop a Spy in a city for a few turns (to gain the "stationary spy" discount, up to and including five (5) turns) and have an E.P. advantage, and you'll find yourself stealing Techs and Poisoning Water like a pro.
Just mouse over your opponant's name in the bottom-right list o' Civs for a quick view of who's got more/less points than you. (Small number first / big number second is good. The reverse is bad.)
