There are two exhibitions currently on in London which are must-sees if you’re in the capital any time soon.

Giacometti’s Pure Presence exhibition at the National Portrait Gallery has had five star reviews already, and on the second day it was open crowds were literally queuing out of the door to buy timed tickets. Despite this the gallery still didn’t feel over-full.

Giacometti is up there in my top two of sculptors, but this is apparently the first exhibition to focus on his portraiture, with more than 60 paintings, sculptures and drawings from international collections. He was fascinated throughout his career about how to evoke a human presence in his work, and how people fitted into space, and this exhibition seeks to explore that.

There are some amazing works from his youth, including his first sculpture, a bronze work of his brother Bruno made when the artist was just 13. Their father was a post-impressionist painter, and early paintings of his family are in a similar style. In later works he experimented with surrealism and developed his own style. The Artist’s Mother of 1927 showed her with a flat face, the features engraved on it.

A whole room focuses on works in which Giacometti’s mother, who lived into her 90s, was the model, being depicted depending on how he was working at that time; in one painting she was the only focus, then in 1950 she was seen smaller and at the end of he room, fitting into the context of household items, related to his interest in existentialism at that time.

His brother Diego worked with him much of his life, and there are several portraits of Diego seated more casually than the female models, and also Diego in a Sweater, a bronze from 1953 showing the jumper as huge, and his brother’s head tiny but sculpted in detail.

As Giacometti became more successful, there were other models from outside the family, including Jean Genet, painted off centre in an oil work from 1954-5.

A room shows photos of Giacometti with his family, his ramshackle studio and with companions including Samuel Beckett. However his interests in people were not limited to the well-known; his wife Annette posed for him many times, and towards the end of his life a model he called Caroline who moved in borderline criminal circles became important, and a series of paintings show him focussing in on her face and then more so her eyes.

It’s a fascinating exhibition which although successful in itself left me pining for more of his sculptures again.

Over at the Royal Academy, Ai Weiwei’s exhibition gained lots of publicity on opening because of the British government’s initial refusal to grant him a visa to enter the country, an act later reversed.

However the exhibition deserves fame on its own account, demonstrating art as an undoubtedly political act and process. There are quite a few items made from wood from dismantled temples from the Qing Dynasty (1644-1911), including an unusable table with two legs up the wall, and a table with pillar coming out of it. Kippe is a large structure made from bits of wood fitted together. Outside, you approach the gallery through wooden trees, made up of parts of different trees fitted together, representing the whole of China.

In one room, names and details line both walls, identifying the thousands of school pupils who died in the 2008 earthquake in Sichuan, in which lots of inadequately-built schools collapsed. Metal rods meant to reinforce them are lined up on the floor in a large installation, and there is a film of the aftermath of the collapse, and photos. Pieces like this have not surprisingly made Ai Weiwei unpopular with the Chinese authorities.

Shanghai Studio is the title of another room, where we are told about how Ai Weiwei was invited to build a new studio in a Shanghai district, and this was done, then it was declared illegal and torn down; bits of brick, cement and ornate wood make up a huge block in the gallery from the destruction, ironically entitled Souvenir from Shanghai.

In another room, the destruction of Chinese historical building and artefacts and the race to destroy more through efforts to boost the economy is explored partly through Dust to Dust, ground dust from china dating back to 5000-3000BC stored in jars.

Fragments is a huge wooden structure you can walk underneath, made from some of the larger, attractive parts of dismantled ancient temples, and in another room the popular Chinese building material marble is turned into models of security cameras used to keep an eye on the artist.

In 2011, Ai Weiwei was held for 81 days as he tried to leave the country, and kept constantly in a room with two guards. He has created six small models of this room, which can be viewed through a window, or by looking down into it from positioned steps. In these he has modelled himself going through the motions of living, as the two guards watch; sleeping, reading, eating, showering and even using the toilet. You get some small idea of how intolerable this must have been.

The final room contains a chandelier made partly of bicycles; a bright, glittering end to an amazing exhibition of works by an artist of great significance today.

* Giacometti, Pure Presence, is on until January 10, and Ai Weiwei until December 13.