Many roads in Italy are based on the roads built by the romans.
The more famous is probably the Via Aurelia: it was built in the third century BC, and it linked Rome to Tuscany, up to Pisa.
In today Italy road nomenclature "Strada statale numero 1" (National road number 1) follow more or less the same itinerary, and it is still called Via Aurelia.
Yes. In Britain many modern roads follow the path of Roman roads, which were rather extensive. You'll often find a winding road turn unerringly straight for a few miles. Many of the Roman roads were of course based on routes which existed before they arrived.
N.b. the map has some inaccuracies (I noticed that Canterbury and Gloucester are out of position) but it gives you an idea.
In spain there is the "via Augusta" the A7 highway follows the old path
The King's road was a major post road between 14th(?) and 18th century in Nordic countries.
Anything that modern would not be noteworthy in Europe. This is an old road that passes close to where I live in Britain. Most of it is not currently an open public road as it is not well sited for current trade routes, but a large part of it remains in use for local agricultural traffic (and legally remains a public road, albeit one usually closed to traffic). There are other old roads in use, but it's the more modern Roman roads which are more important. Most trunk roads in England used the Roman routes until the building of the motorways in the 1950's to 70's. However don't think that a Roman road necessarily became a major modern road - as with the old roads, most of them degenerated to intermittent country roads and tracks due to changing centres of population. So for instance Silchester, an Iron age oppidum and latterly a Roman centre, to the south of present-day Reading, was abandoned. The road network around it is easily seen on a modern map, but the once-major roads are now just country lanes or ploughed under fields.
Accross Northern France and Belgium there is a full network of "Chaussée Brunehaut". Those were ancient Roman ways rehabilitated by Brunehault, a ruler of the place during Merovingian times. They tend to be very straight as Roman ways were.
Interestingly, the straight line from Boulogne (on the coast of France) to Koeln (Germany) used to be a Roman way that pretty much marks the divide between Romance and Germanic languages. It is still somewhat clear in the toponymy (just google maps to see the pattern), even though population moves have moved the frontier a tad up North and West since the low middle ages.
In a nutshell, the Roman Via linked Roman outposts on this frontier land.
I know its not western Europe, but the Grand trunk road in northern india has certainly existed since Mughal times 16th century, and is supposed to have been first built under the Mauryas (3rd century BCE)
