These have been a total highlight of the trip. We saw Monte Albán, Palenque and Tulúm, and here are my impressions. I hope I haven't made too many mistakes with my historical detail. Forgive me if I have.

Monte Albán is a Zapotec ruin, right outside Oaxaca. It dates from 200 to 800 AD.

The Zapotec people preceded the Maya, who built the most famous ruins in Mesoamerica (Palenque, Chichén Itzá and Tikal). The Zapotecs were pretty smart - they invented the first Mexican calendar aroun 500 BC.

They settled in a valley (there were 20,000 people living in the city by the first year of the common era), and decided to build their ceremonial centre on top of a tall hill. They had no knowledge of metals, now wheel, and no domesticated animals. To flatten the hilltop, they had to break stone against stone by hand and carry them away. They had the plan for the centre all organised before they commenced. Archeologists think it took at least three or four generations just to flatten the hilltop. I was amazed to hear this. What kind of faith must people have had in their leaders and gods to work so hard on a project that even their grandchildren would never see completed? And how about the leaders themselves? They stuck to the plan with a sense of focus that seems astonishing.

Anyway, the ruin is very impressive, although simple when compared to the detail you can find in other places, like Palenque. Our guide told us that the Zapotecs practised human sacrifice only modestly - maybe a few times a year. To be sacrificed was an honour, and the ritual took place when things were going wrong. Apparently the Zapotecs didn't consider their gods to be infallible or immortal, and each god was related to a part of nature - the sun, the rain, or the wind. So if the crops were going badly for lack of water, someone would be sacrificed so he could go to heaven and help the rain god, who was clearly unwell.

As one civilisation followed another, human sacrifice was practised with less religious purpose, and more political purpose, as a tool of intimidation, said our guide. (The Mixtecs were pretty keen on it, but nobody had a bloodlust like those Aztecs. The idea of honour faded from the ritual, and often slaves and prisoners of war would be sacrificed, our guide told us. The Aztecs also had a particularly gruesome method, cutting out the victim's heart while he was still alive, just like that scene in the Indiana Jones movie).

A huge storm gathered as we walked aroun the site, and it was easy to imagine how impressive it must have felt to be here during important rituals, overlooking the village and the land all around. We could see the rain moving across the valley like a thick shadow.

We traveled to Palenque from the delightful town of San Cristóbal de las Casas, in Chiapas. This was where the 1994 Zapatista rebellion had its moment of triumph, and now the local craftspeople sell little Comandante Marcos souvenirs - t-shirts and keyrings. He's become a sort of Ché.

Anyway, we loved Palenque. It is a Mayan ruin in the middle of the jungle. It's a very different part of the country - much more tropical and much, much hotter. The pyramides appear out of the jungle, and when you climb some of the higher ones, you acn look out onto the vast Yacatan plain. It's most impressive.

The ruins here are much richer than the ones at Monte Albán in terms of detail. There are glyphs on many of the buidings, and sculptures depicting gods and kings. They know that Palenque was ruled by a very successful dynasty, and enjoyed a lengthy golden age from around 600 to 800 AD, and was abandoned around 900 AD. The royal family held onto power by marrying within itself, so lots of the pictures show people with deformities.

Unlike at Copán, in Honduras, another Mayan ruin, the glyphs at Palenque can't be deciphered. But they're beautiful, and the city must have looked incredible when it was at its most glorious. All the ruins are bare stone, now, but when they were completed, they were covered in stucco and painted in bright blues and reds. There were some terrific examples of the Mayan arch, a triangular arch that is found in most of the late Mayan sites and is very graceful. The buildings here all have roof-combs, too - a sort of stone lattice-work detail that gives the heavy buildings a kind of lifting feeling.

Some of the surrounding hills near the archeological site still have temples on them, buried under the jungle. The archeologists will need lots more money if they are to go on exploring the city.

As we wandered around, nearly fainting from the heat with our friendly guide, Victor, we could hear the monkeys screaming in the jungle, and we saw a Tucan fly past. Victor said we were lucky.

Our last ruin, Tulúm, is appealing mostly for its position - the temple is perched on a cliff overlooking the Caribbean. It's a walled ceremonial centre that is one of the latest Mayan cities. As in other cases, the ordinary people lived outside the part of the town that was built from stone, and only came inside for special occasions. Tulúm is a small ruin, and we walked around pretty quickly. It was hot and we were keen to jump into the sea.

I think all the great pre-hispanic cities in Mexico and Guatemala, except for the magnificent Aztec city of Tenochtitlán, now Mexico City, were abandoned by the time the Spanish arrived - Mónte Albán started collapsing around 700 AD, and Palenque was abandoned by 900 AD, with the other Mayan sites in Mesoamerica (Copán and Tikál) collapsing around the same time. The cities quickly became hidden by the creeping jungle and couldn't be seen. Nobody knows exactly why these great Mayan cities, or Teotihuacán, the ruin just outside Mexico City, were abandoned. Maybe famine - some people say the cities grew too fast and the food supply couldn't keep up. Maybe it was also because they had a lot of wars among themselves. It's a mystery.