Explore the fascinating world of multivariate totally positive of order two (MTP2) probability distributions in this comprehensive colloquium talk. Delve into the intriguing properties of MTP2 distributions and their applications in various fields, from ferromagnetism to phylogenetics and finance. Learn about conditional independence, graphical models, and how maximum likelihood estimation for MTP2 exponential families leads to sparsity in Gaussian or Ising models without tuning parameters. Discover the challenges of non-parametric density estimation under MTP2 and its piecewise linear function solution. Gain insights into practical applications, including covariance matrix estimation for portfolio selection and learning gene regulatory networks. The talk covers motivation, theoretical foundations, examples, and concludes with an engaging Q&A session.