This series of lectures will be devoted to applications of convex geometry to problems of high-dimensional statistics. The basic notion is that of a polyhedral convex cone in the \(d\)-dimensional Euclidean space. It is defined as an intersection of finitely many half-spaces whose bounding hyperplanes pass through the origin. In other words, a polyhedral convex cone is the set of solutions to a system of finitely many linear homogeneous inequalities. Alternatively, a polyhedral cone can be defined as the set of positive linear combinations \(\lambda_1 x_1+\ldots+\lambda_n x_n\) of a finite collection of vectors \(x_1,\ldots, x_n\) in \(\mathbb R^d\) with non-negative coefficients \(\lambda_1,\ldots,\lambda_n\geq 0\). By intersecting a polyhedral cone with the unit sphere centered at the origin we obtain what is called a spherical polytope. To each polyhedral cone \(C\subset \mathbb R^d\) one can associate the vector of conic intrinsic volumes \(v_0(C), v_1(C),\ldots,v_d(C)\). These conic intrinsic volumes are the spherical analogue of the usual intrinsic volumes studied in convex geometry. We shall review the definition and main properties of conic intrinsic volumes and describe their applications to some problems of stochastic geometry, for example to counting faces of randomly projected cubes, simplices and other polytopes. These problems, studied in the works of Donoho and Tanner, have applications to high-dimensional statistics, signal processing and compressed sensing which we shall also explain. On the other hand, applications of conic intrinsic volumes to phase transitions in convex optimization problems with random data have been discovered in the work of Amelunxen, Lotz, McCoy, Tropp. We shall explain these applications and, if time permits, also the interconnections between conic integral geometry and various other topics such as random matrices, hyperplane arrangements and the classical Sparre Andersen arcsine laws for random walks.