Perpetual motion would be a wonderful thing, if only it were possible - being able to set some machine going and then it would power itself and just carry on & on without end. Free energy from nothing! Which is, of course, why it is impossible - but this wasn't provable until relatively recently. Discussing the search for, and disproof of, perpetual motion on In Our Time were Ruth Gregory (Durham University), Frank Close (University of Oxford) and Steven Bramwell (University College London).
Before the modern understanding of physics there didn't seem to be any reason why perpetual motion should necessarily be impossible. In the Aristotelian view of the universe the stars were in perpetual motion in the heavens - so there must surely be some way to replicate this on earth with earthly machinery. This wasn't (solely) the province of charlatans - people like Leonardo da Vinci, Robert Boyle and Gottfried Leibniz were all involved in attempts to design machines that could power themselves forever. Various approaches were tried - like trying to design a waterwheel that not only ground corn but also pumped the water back uphill so it could flow down again. Or a bottle that siphoned liquid out of itself in order to refill itself. Or some sort of machine that was constantly over-balancing - like an Escher drawing of a waterwheel with buckets labelled 9 travelling down one side, and when they reach the bottom they flip round to now read 6 so they're lighter. Which works beautifully in the illusory world of Escher's art but rather less so in our mundane reality. As well as people genuinely trying to investigate the possibility there were also those who claimed to have achieved success - normally with machines that conveniently couldn't be inspected to expose their charlatanry.
Once physicists started to gain a greater understanding of how the universe worked it became clear that perpetual motion machines were fundamentally impossible. All proposed perpetual motion machines violate either the first or second law of thermodynamics. Before moving on to explain how these laws affect perpetual motion machines they digressed slightly to explain some of the background to the formulation of the laws of thermodynamics. First they gave us the technical meaning of the word "work" in a physics context - important in understanding the rest of the discussion. Work is the energy that is applied to do something. For instance if you want to move something then work = force x distance. Or if you're heating something up then work refers to the energy you need to cause the temperature change. Experiments by Joule were key to showing a link between heat and energy. Before his work the prevailing theory was that heat was a thing (called calor) that could be transferred between objects - so a fire heated a pot because calor was transferred from flames to pot. But Joule showed that you could generate heat using energy, and later it was realised that heat was a form of energy. Reception of his work at the time (the mid-19th Century) was mixed - the temperature changes he was study were very small and not everyone believed it was possibly to accurately detect them.
The First Law of Thermodynamics is that energy must be conserved in a closed system. I.e. you don't get something for nothing. When work is done it all turns into motion or heat or some other form of energy. Many perpetual motion machines violate this law, and they are termed "perpetual motion machines of the first kind". An example of this is a waterwheel that both grinds corn and pumps the water back up to the top to start over again. In order to pump all the water back up you need to use just as much energy as it generated for you on the way down - so there none left over for your corn grinding, even if your machine is perfectly efficient (see below).
The Second Law of Thermodynamics is that entropy always increases or remains the same, it never decreases. Gregory used the example of a room that's either tidy (a single ordered state) or untidy (many possible disordered states). In order to move from disordered to ordered you need to do work, otherwise over time the random chance will move objects from their positions in the room and it will become more disordered. The Second Law of Thermodynamics is associated with time - it provides directionality to the universe, if things are getting more disordered then they're moving forwards. Perpetual motion machines which violate this law are categorised as "perpetual motion machines of the second kind".
Another way that perpetual motion machines can violate the laws of physics is by being too efficient. I touched on that above - in the real world no machine operates without losing some energy (generally in the form of heat due to friction). And so even if you aren't trying to do anything useful with the energy other than keep the machine moving you'll still fail to achieve perpetual motion as you won't have quite enough energy to return to the starting point.
So perpetual motion is impossible, as it would violate the laws of physics. There are some loopholes at the quantum level (aren't there always?). Implications of the Heisenberg Uncertainty Principle mean that it's possible to "borrow" energy temporarily from the future, which means the First Law of Thermodynamics doesn't quite apply. But at the macro level these laws are inviolable and perpetual motion is impossible. They finished by saying that if a way to make a perpetual motion machine work was found then it wouldn't just be a case of minor tweaks to physics-as-we-understand-it. Instead it would require a re-writing of pretty much all science we've ever conceptualised - the laws of thermodynamics are that fundamental to our understanding of the universe.