This volume is based on lectures presented at the AMS Special Session on Algebraic Methods in Statistics and Probability--held March 27-29, 2009, at the University of Illinois at Urbana-Champaign--and on contributed articles solicited for this volume. A decade after the publication of Contemporary Mathematics Vol. 287, the present volume demonstrates the consolidation of important areas, such as algebraic statistics, computational commutative algebra, and deeper aspects of graphical models. In statistics, this volume includes, among others, new results and applications in cubic regression models for mixture experiments, multidimensional Fourier regression experiments, polynomial characterizations of weakly invariant designs, toric and mixture models for the diagonal-effect in two-way contingency tables, topological methods for multivariate statistics, structural results for the Dirichlet distributions, inequalities for partial regression coefficients, graphical models for binary random variables, conditional independence and its relation to sub-determinants covariance matrices, connectivity of binary tables, kernel smoothing methods for partially ranked data, Fourier analysis over the dihedral groups, properties of square non-symmetric matrices, and Wishart distributions over symmetric cones. In probability, this volume includes new results related to discrete-time semi Markov processes, weak convergence of convolution products in semigroups, Markov bases for directed random graph models, functional analysis in Hardy spaces, and the Hewitt-Savage zero-one law. Table of Contents: S. A. Andersson and T. Klein -- Kiefer-complete classes of designs for cubic mixture models; V. S. Barbu and N. Limnios -- Some algebraic methods in semi-Markov chains; R. A. Bates, H. Maruri-Aguilar, E. Riccomagno, R. Schwabe, and H. P. Wynn -- Self-avoiding generating sequences for Fourier lattice designs; F. Bertrand -- Weakly invariant designs, rotatable designs and polynomial designs; C. Bocci, E. Carlini, and F. Rapallo -- Geometry of diagonal-effect models for contingency tables; P. Bubenik, G. Carlsson, P. T. Kim, and Z.-M. Luo -- Statistical topology via Morse theory persistence and nonparametric estimation; G. Budzban and G. Hognas -- Convolution products of probability measures on a compact semigroup with applications to random measures; S. Chakraborty and A. Mukherjea -- Completely simple semigroups of real $d\times d$ matrices and recurrent random walks; W.-Y. Chang, R. D. Gupta, and D. S. P. Richards -- Structural properties of the generalized Dirichlet distributions; S. Chaudhuri and G. L. Tan -- On qualitative comparison of partial regression coefficients for Gaussian graphical Markov models; M. A. Cueto, J. Morton, and B. Sturmfels -- Geometry of the restricted Boltzmann machine; M. Drton and H. Xiao -- Smoothness of Gaussian conditional independence models; W. Ehm -- Projections on invariant subspaces; S. M. Evans -- A zero-one law for linear transformations of Levy noise; H. Hara and A. Takemura -- Connecting tables with zero-one entries by a subset of a Markov basis; K. Khare and B. Rajaratnam -- Covariance trees and Wishart distributions on cones; P. Kidwell and G. Lebanon -- A kernel smoothing approach to censored preference data; M. S. Massa and S. L. Lauritzen -- Combining statistical models; S. Petrovi?, A. Rinaldo, and S. E. Fienberg -- Algebraic statistics for a directed random graph model with reciprocation; G. Pistone and M. P. Rogantin -- Regular fractions and indicator polynomials; M. A. G. Viana -- Dihedral Fourier analysis; T. von Rosen and D. Von Rosen -- On a class of singular nonsymmetric matrices with nonnegative integer spectra; A. S. Yasamin -- Some hypothesis tests for Wishart models on symmetric cones. (CONM/516)