All Week
BP PORTRAIT AWARD: The annual free showcase of the most talented portraitists is back at the National Portrait Gallery. This year the competition had a record-breaking 2,748 entries — subjects include famous people, nudes, family portraits and photo-realism. Free entry, until 20 September
CITY OF LONDON FESTIVAL: The City of London is crammed with music, comedy, cabaret, performance, film and more for City of London Festival. Some events free, see programme for details, until 10 July
TRAVELLING ARCHIVE: The Travelling Archive is an exhibition of sounds, texts and images, all taken from a research, documentation and dissemination project in Bangladesh, West Bengal and other adjoining parts of eastern India, as well as Bengali neighbourhoods in London. See the exhibition at Rich Mix. Free, just turn up, until 5 July
MORE LONDON FREE FESTIVAL: From now until September, The Scoop at More London Riverside will be taken over by theatre, music, dance and film, as the More London Free Festival takes place. This month, the focus is on music. Take a look at the calendar to see what’s on this week.
ILLUSTRATION AWARDS: See this year’s entries to the V&A Illustration Awards at the museum. Works consist of book, editorial and student illustration. Free, just turn up, until 2 August
CHRISTINA BROOM: Museum of London Docklands’s new exhibition is all about the UK’s first female press photographer, Christina Broom, a self-taught novice who turned photography into a business venture to support her family. Free, just turn up, until 1 November
FREE BURGER: Prime Burger launches in St Pancras International this week and to celebrate, they're giving away 100 free gourmet burgers (including the UK's first ever truffle burger) every day. If you're one of the first 100 people through the door of Prime Burger between 1pm and 2pm, with a voucher (available here), the burger's yours. 100 per day, 1pm-2pm, 29 June-3 July
Monday 29 June
THEATRE PLATFORM: The National Theatre's Jocelyn Herbert Lecture features German director Walter Asmus reflecting on his working relationship with Samuel Beckett. £4/£3, prebook, 6pm
TWILIGHT SCIENCE: The Royal Society holds a science evening with a preview of the Summer Science Exhibition. Meet science experts and hear them discuss their work — topics range from lasers to sand dunes to Nerf guns. Adults only. Free, 6pm-10pm
COMEDY: Previews at The Camden Head from Chris Betts, who’s been watching us all when we’re drinking, and John Hastings, who’s considering parental relationships. £5, just turn up, 8pm
Tuesday 30 June
PHOTOGRAPHY EXHIBITION: Beetles + Huxley Gallery in Mayfair hosts the first major UK exhibition of photographer Paulette Tavormina. It largely consists of shots of fruit and flowers, inspired by still life paintings. Free, 30 June-25 July
HISTORY TOUR: Morden Hall Park has free guided tours, led by knowledgeable tour guides. Walk a gentle tour of the park and discover stories you never knew. Free, prebook, 11.30am/1pm
SIR CHRISTOPHER WREN: Gresham College is at Barnard's Inn Hall for a lecture about how Sir Christopher Wren went about rebuilding over 50 churches in the decades after the Great Fire, and how economic, political, religious and aesthetic factors shaped the appearance of these much-loved buildings. Free, just turn up, 6pm
RAILWAY HISTORY: James Attlee and Will Gompertz are at Stanfords in Covent Garden to discuss the history of the Great Western Line, engineered in the 19th century by Isambard Kingdom Brunel. £3, prebook, 6.30pm
SHORTS ON TAP: Shorts On Tap is at Cafe 1001 in Brick Lane for a screening of seven short documentaries, curated in collaboration with the London Documentary Network. £5, prebook, 7pm
SPRING HEELED JACK: London Fortean Society is at Conway Hall in Holborn for a talk about Spring Heeled Jack, a mythological creature from Victorian London. £5, prebook, 7.30pm
FREE COMEDY: Get down to the Star of Kings in King’s Cross for sketch, character and stand-up comedy from Twins, Stuart Laws, Matthew Highton, Lou Sanders and Natasia Demetriou and Ellie White. Free, just turn up, 7.30pm
Wednesday 1 July
VARIATIONS: A new exhibition opens in the Gallery at Oxo Tower today, bringing together the work of three established London-based artists, Giuseppe D’Innella, Jayson Lilley and Anna Masters and showcasing new and previously unseen work. Works range from painting and screen printing to mixed-media sculpture. Free, just turn up, 1-5 July
LEYTONSTONE FESTIVAL: Leytonstone Festival starts today, with plenty of free events including family theatre workshops, film festivals, open mic nights and concerts. See full programme. 1-19 July
LAMBETH PALACE: Lambeth Palace Gardens are open to the public on the first Monday of each month, with teas and coffees available to buy. £4, 12pm-3pm
LONDON'S BURNING: Radical bookseller Housmans's London's Burning series begins today with a talk by Matthew Beaumont, who discusses his London book Night Walking. £3, prebook, 7pm
GLOBAL SHORTS: Head to New Cross Learning Centre for a free screening of 16 short films from around the world curated by Joel Blackhage, from Chinese animation to Kenyan dance. Free, 7pm
DASH ARTS: Russian musician and producer Anton Maskeliade is at Dash Arts at Rich Mix. He is one of the first artists in the world to create live electronica with gesture control technology leap motion. He'll be talking about the technique and performing a live set. Free, just turn up, 7.30pm
DISASTER FILM CLUB: Disaster Films, a film club dedicated to films with exceptional soundtracks, screens Into Eternity: A Film For The Future. Takes place at Hackney Picturehouse. £3, prebook, 8pm
Thursday 2 July
PHOTOGRAPHING THE HOME FRONT: London Metropolitan Archives in Finsbury offers a chance to explore photographs of London during World War Two, suffering the effects of enemy fire, but continuing to operate, regardless. Free, prebook, 3pm
THEATRE PLATFORM: At National Theatre, writer, director and performer Tim Crouch talks to Emily McLaughlin about the production, An Oak Tree. Free, prebook, 4.30pm
OLYMPIC STADIUM: This talk, at The Institution of Structural Engineers near Old Street, covers the design and erection engineering behind the innovative gravity-stressed cable-net roof which will be put onto the Olympic Stadium when it becomes a football stadium. Free, prebook, 5.30pm
OUT OF THE ASHES: The British Academy hosts a talk by Professor Sir Ian Kershaw about Europe’s rebirth after the Second World War. Free, just turn up, 6pm
LITERARY NIGHT: The Working Men's College in Camden has an evening of author readings, hosted by Lucy Popescu. Authors include Pele Cox, Carol Lee and Hannah Lowe. Free, prebook, 7pm
Friday 3 July
AFRICA WRITES: Africa Writes is the Royal African Society’s annual literature festival, taking place at The British Library. Over the course of the weekend, there are several free events going on, including a chance to meet publishers, a book fair and more. 3-5 July
FRIDAY LATE: Wellcome Collection on Euston Road is open late for a night of games and play for grown-ups, from intriguing new digital games to traditional parlour favourites as well as discussions and experiences based around the different ways we can play. Free, just turn up, 7pm-11pm
LONDON DREAMTIME: Storytelling experts London Dreamtime are joined by a special guest, storyteller Christine Cooper, for some myths and legends in a secluded and muddy spot on the foreshore of Thames. Stories at twilight, mudlarking first for those who fancy it. Location revealed on booking. £3, prebook, 8.30pm
Saturday 4 July
BANK OPEN DAY: As part of City of London Festival, Bank of England has a rare open day. Take a guided tour of parts of the building, including the Court Room and the Committee Room, which is where the Bank's Monetary Policy Committee meets once a month to set Bank Rate. Free, get there early as it's bound to be popular, from 9.30am
STREET CELEBRATION: An open air street celebration launches Sydenham Arts Festival today. Expect live music, dance and street performances along the high street and beyond. See full festival programme. Free, just turn up, 11am-4pm
LEYTONSTONE FESTIVAL: The official launch party of Leytonstone Festival takes place today. The fun starts at St John’s with live music, singers, dancers, food, exhibitions, children’s activities and puppet shows. There's also a parade, and the start of the Leytonstone Arts Trail event. Free, just turn up, 11am-5pm
BERMONDSEY SONG WALK: Join Nigel Of Bermondsey in a walk from Borough to Rotherhithe, with songs performed at the locations which inspired them and plenty of river views. Free, prebook, 4pm-7pm
LIVE MUSIC: Alternative rock band SALT headline at The Islington in Angel, where they'll be debuting a new song. Entry is only £3, and that includes a shot— what's not to like? £3, just turn up, 7pm
Sunday 5 July
SUMMER STREETS: It's the first of Regent Street's Summer Streets events, when the busy shopping thoroughfare is closed to traffic, and events and entertainment take place throughout the day. Free, just turn up, all day
HYDE BARK: The Mayhew Animal Home takes over The Cockpit in Hyde Park for a fundraising day with a dog show, agility displays, vet demonstrations, pet behaviour advice, face painting and lots more. Free entry, just turn up, 10am-5pm
VINTAGE FURNITURE FLEA: Head to York Hall in Bethnal Green to get your hands on retro atomic-prints, ornaments, cushions, crockery and glassware, dating back to the 50s at the East London Vintage Furniture Flea. £3/£2, 10.30am-4.30pm
DRAGON BOATING: Ever fancied dragon boating? Have a go at the Typhoon Dragon Boat Club Summer Open Day. Meet team members, learn the basic paddling technique and more. Free entry, just turn up, 11am
MUSIC ON THE LAWN: Forty Hall & Estate in Enfield hosts an afternoon of live music. Bring a picnic to tuck into while you listen to jazz, blues and soul. Free, just turn up, 12pm-6pm
ACOUSTIC SUNDAYS: Head to St Peter's Church in Haggerston for the Midsummer Gala, an afternoon of acoustic music, open mic sessions and local artists. Food and drink available to buy. Free, just turn up, 2.30pm
Check out our map of free things to do in London for more frugal fun.