Baku, Absheron & Gobustan, 4 Nights

Heydar Aliyev airport sits on the Absheron peninsula east of the city, and the drive in runs along the Caspian shore. Check-in is usually from mid-afternoon. Once rested, go up to Highland Park, the terrace above the bay where the Alley of Martyrs remembers those killed in January 1990. From here the whole crescent of Baku is laid out, and at dusk the three Flame Towers behind you switch on their moving LED display. It is the best first view the city offers, and it costs nothing more than the ride up.

A morning on foot inside Icheri Sheher, the walled town inscribed by UNESCO in 2000. The Palace of the Shirvanshahs was the court of the fifteenth-century rulers of Shirvan; the Maiden Tower, older and still argued over, stands at the edge of what was once the shoreline. Caravanserais now hold carpet sellers and teahouses. After lunch, Zaha Hadid's Heydar Aliyev Center, a white building that rises out of its plaza like a folded sheet, then back to the seafront for the boulevard, the Carpet Museum shaped like a rolled rug, and Fountains Square in the evening.

East to Surakhani and Ateshgah, a walled fire temple built in the seventeenth and eighteenth centuries around a central altar. Its cells carry inscriptions in Devanagari and Gurmukhi as well as Persian, left by Indian merchants and ascetics who travelled the caravan routes; the original gas vent failed in 1969 and the flame is now piped. Then north to Yanardag, where a seam of natural gas burns along a low sandstone bank. The flames show best late in the afternoon, which is why this day runs east first and ends on the hillside.

South along the coast road, about 60 km, to Gobustan, whose boulders carry thousands of engravings of hunters, boats, dancers and animals; the site was inscribed by UNESCO in 2007, and the museum at its foot explains what is being looked at before you walk up. The mud volcanoes lie a few kilometres further on, over a rough track that local drivers cover in old Ladas. On the way back, a stop at the Bibi-Heybat mosque, rebuilt in the 1990s on the site of a thirteenth-century mosque destroyed in 1936.

Breakfast, and time to finish the shopping that the week has lined up: Azerbaijani pakhlava layered with walnuts and saffron, tea in pear-shaped armudu glasses, or a carpet chosen earlier in the Old City, provided the paperwork for its export was sorted when it was bought. Checkout is usually at noon, and the transfer back along the shore to the airport takes about half an hour outside the evening rush.