This year's 18.177 course is a selective introduction to statistical physics, probability, and field theory. The course is structured around four unsolved problems, listed and explained below. These problems have received a fair amount of attention. Solutions to special cases of Problem 2 have led to two recent Fields medals, for example, and the Clay Institute has placed a million dollar bounty on Problem 4. I chose these particular problems because thinking about them gives us an excuse to internalize fundamental concepts that we would have wanted to learn anyway (ergodic theory, zero-one laws, Brownian motion, FKG inequalities, Ito calculus, the Gaussian free field and other random distributions, Schramm-Loewner evolution, etc.) and to study some exciting new results (Smirnov's recent work on percolation and the Ising model, for example). The existing literature on the problems is already quite rich, and the problems build on each other in a coherent way. For each of the four units, students taking the course for credit will be required to complete a problem set and provide a short summary of their own attempts to make progress (however limited) on the problem. It will be great if publishable research emerges from this course, but for an A grade it will suffice to give a good short exposition (couple of pages) of an existing result and an explanation of how one gets stuck when trying to apply it to the big problem. The progress summaries can be completed as group projects in collaboration with other students.