What is reactivity really, and can it be quantified? I understand the concept of a reactive species on a sort of vague and intuitive level: a reactive species is one that has lots of reactions that it can participate in. These tend to have large kinetic constants, be thermodynamically downhill and often have products that are also reactive, leading to knock-on effects. But reactivity isn't the same as energy, and it's possible to have highly reactive species with relatively low free energies and vice versa.

I'm interested in whether there's any kind of formal theory of reactivity, and/or a quantitative definition of the concept. For example, we're often told that free radicals are reactive because they have an unpaired electron and they "want" to fill their outer shell. This makes perfect sense on an intuitive level, but I'd like to know if it can be taken further. Is there some way we can write down an equation that will tell us how reactive a species will be in a given environment, or is there some way we can measure a species' reactivity empirically and use it to make predictions?

To clarify, I'm not looking for quantum-level predictions, although I will look into the theories mentioned in Sam and Greg's answers. Rather, I'm looking for something a bit more heuristic that will help me get a handle on what reactivity really is. In particular, what does it really mean to say that one species is more or less reactive than another? Can we put a number to it and say that a species has a reactivity of 8.2 on some meaningful scale?

ron's answer is a start. It surely must be the case the reactivity relates to the energy landscape. But ron's answer deals with a one-dimensional landscape, corresponding to a single reaction. Reactivity is a concept that involves many reactions, and so a satisfying theory of reactivity would have to involve the properties of a massively multidimensional landscape. If there is anything that has been written about it from that point of view I would be very happy to read it.
I think the term "reactivity" is often used to denote kinetic lability, a proclivity to react.  Let's look at the two reaction coordinates shown below.  They both represent the interconversion of molecules A and B through intermediate I.



Note that the potential well that I exists in is much deeper in the drawing on the left than the drawing on the right.  Therefor the activation energy for I to escape will be much larger for the intermediate on the left than the intermediate on the right.  The intermediate on the right would be said to be more reactive (have a shorter lifetime) than the intermediate on the left, because it requires less energy for it to escape.

An experimentalist can infer the depth of the well through experiments.  Through such experiments the experimentalist learns that free radicals, highly strained hydrocarbons, carbenes, etc. exist in potential wells that must be very shallow because they react so readily.  Using this empirical data the experimentalist can then infer how similar compounds are likely to react. A computational chemist can infer the same thing by calculating the profile of the reaction coordinate for the various reactions under investigation by the experimentalist, or as is becoming increasingly common the computational chemist can predict what should be and let the experimentalist confirm those predictions.