Just bought house with hairline diagonal crack from window

From the pictures it looks like the cracks may only be in the stucco, and the underlying structural wall may be unaffected. It's hard to tell without actually inspecting the wall, but my guess is that these cracks are only skin deep. However, you will want to treat them to prevent them from spreading.

Here is a good article about Cracks in stucco.

Here are some excerpts from the article.

All buildings are subjected to a variety of stresses in various forms and this stress is often transferred to the plaster stucco membrane.

Stucco is not and never was intended to be a structural material.

Cement stucco is a relatively thin, brittle shell over a framed structure and can crack if the stresses are transferred and overcome the strength of the plaster membrane.

SHRINKAGE STRESS

All Portland cement shrinks as it cures. Controlling and minimizing that shrinkage is critical to minimizing cracks

Shrinkage cracking tends to have a map-like cracking pattern.

OUTSIDE SOURCES

Structural loads, lateral force racking, thermal changes, wind, lumber shrinkage and seismic events are all examples of outside sources of stress.

HOW HARMFUL IS A HAIRLINE CRACK?

a hairline crack does not automatically mean the stucco is faulty, failing or will not function for the service life of the building.

MAKING REPAIRS

Once the decision is made to make repairs, there are several options.

Option 1: Treat the joint with the same finish material.

Option 2: Paint the wall or ceiling after treating the crack with an acrylic paint or paint designed for stucco.

Option 3: An elastomeric paint coating.

Option 4: Recoat the wall from architectural break to architectural break with the same finish material.

Option 5: A new basecoat and mesh over the wall area.

The important thing about cracks is not necessarily their absolute number or size - though if you've got a particularly long one it can be a sign of problems, but whether the number of cracks is growing and/or the existing cracks are getting longer and/or wider.

A new crack should be monitored over a few days/weeks to see if it gets longer and/or wider and how quickly. If it's changing quickly then call in an expert sooner rather than later.

Who's an "expert" in these cases? Well for less serious cracks a builder should be able to advise you. For larger, wider, deeper cracks I'd go for a surveyor as they'd be more experienced in this area and should look harder for the underlying cause. A decent builder should also recommend a surveyor if they think things are particularly bad.