Plaid Cymru's Leanne Wood, Professor John Curtice and Labour's Alun Davies MS analyse the Caerphilly by-election result. Ann Davies MP tells us how coal tips are still not safe, 59 years after the Aberfan disaster. And Lord Peter Hain talks about his new book, Liberation and Corruption: Why Freedom Movements Fail.

Sam Bennett and Claire Vaughan review the papers.

Vaughan Roderick with political news, discussions and analysis, plus a review of the Sunday papers.

The former Labour MP and Foreign and Commonwealth Office minister for the Middle East Dr Kim Howells discusses the chances of the on-going peace process in the Middle East delivering for both the Palestinian and Israeli people. Plaid Cymru's leader at Westminster, Liz Saville Roberts MP looks back at her party's conference - the last before next May's Senedd elections. Paris-based commentator Nabila Ramdani assesses the prospects of France's new PM - who is the same as the old one - bringing political stability to the Republic. And the the UK's Independent Anti-Slavery Commissioner, Eleanor Lyons looks back at 10 years of the Modern Slavery Act

Neuroscientist Dr Emma Yhnell and Conservative Senedd candidate Adam Robinson review the papers.

Haaretz columnist Gideon Levy and physiotherapist Rachel Moses OBE discuss two years of conflict in Gaza. Children's Commissioner Rocio Cifuentes is asking all the parties to commit to five priorities to improve children's lives in Wales, Darren Millar MS is in Manchester for the Conservatives' UK Conference. Former head of the First Minister's office explains how important EU funding was to Wales and Professor Laura McAllister tells us why there's been such a rise in the number of podcasts on Welsh politics.

Colegau Cymru's Rachel Cable and the Vale of Glamorgan's Council Leader Lis Burnett review the papers.

Deputy First Minister Huw Irranca-Davies joins us from Labour's conference in Liverpool. Jane Frankland CEO of Knewstart discusses cyber security. Emma Pinchbeck, CEO of the Climate Change Committee explains what went well and what went badly after the last blast furnace closed in Port Talbot. Mark Serwotka joins us to tell us all about new left-leaning 'Your Party' in Wales and Sir Deian Hopkin remembers the 1968 Caerphilly by-election.

Tessa Marshall and Karl Davies review the papers.

After Jeremy Miles' shock announcement that he's standing down, we have guardian columnist, podcaster and newsletter writer Will Hayward mulling over Labour's situation in Wales. Former UN Ambassador Sir Emyr Jones Parry and BBC correspondent Wyre Davies recently returned from the Middle East discuss the latest in the Gaza-Israel war. After Jimmy Kimmel's show was cancelled in the US, journalist Cai Wilshaw discusses free speech in America. It's the Liberal Democrat's UK conference this week and Welsh leader Jane Dodds joins us fresh from Bournemouth. And Aled Eirug talks about his new biography: Dafydd Elis Thomas: Nation Builder.

The RCN's Helen Whyley and Plaid Cymru's Baroness Carmen Smith review the papers.

Journalist and author of the unauthorised biography of Peter Mandelson, Paul Routledge explains how the Ambassador's sacking was inevitable.

Professor Kate Thirlaway and Cllr Calum Davies review the papers

Kitty Donaldson, Chief Political Commentator at The i Paper discusses Deputy Prime Minister Angela Rayner's resignation and the reshuffle that followed, newly elected leader of the Greens Zack Polanski has been to Cardiff and talks about his hopes for his party, Fin Wilson from Impact Wales tells us how literacy could be improved in Wales and Caroline Jones joins us from Reform's conference in Birmingham.

ERS Director Jess Blair and Cardiff Council Leader Huw Thomas review the Sunday papers.

Vaughan Roderick looks back at highlights from Sunday Supplement over the past few months, including a row over rail funding, the future of Ukraine and Syria, the passing of Pope Francis, and the 'Great Bibles' going on display at the National Library of Wales.

Interviews with Ben Lake MP, Ukrainian journalist and refugee Kateryna Gorodnycha, Children's Commissioner Rocio Cifuentes, Syrian doctor Mohammad Alhadj Ali, Delyth Jewell MS and Lord Don Touhig, Professor Elin Haf Gruffydd Jones and the National Library's Timothy Cutts.

Vaughan Roderick presents the highlights of the past few months. Alun Davies MS discusses the effect of welfare changes on his Blaenau Gwent constituents; Shereen Williams and Matt Kilcoyne have different views on the new electoral boundaries' names for the next Senedd election; Peter Kellner and Professor Matt Qvortrup discuss the benefits and drawbacks of referenda; Dr Anne Harrington remembers the bombings of Hiroshima and Nagasaki; and Judith Mackrell talks about sibling artists Gwen and Augustus John.