This text is an introduction and overview of the study of singularities of differentiable maps, written by J. M. Boardman. It discusses the classification of singularities of smooth maps between manifolds, focusing on the singularity subsets and their properties. The author introduces the concept of jet spaces and uses them to construct subsets of the jet space that are independent of any specific map. These subsets induce subsets of the domain manifold, and the main theorem states that these subsets are submanifolds in the jet space, implying that for "sufficiently good" maps, the induced subsets are submanifolds of the domain manifold. The codimensions of these subsets are computed, and it is shown that they coincide with those obtained by a naive geometric approach. The infinite jet space and its associated total tangent bundle are used extensively. The text also discusses the inductive procedure for constructing the singularity sets and intrinsic derivatives, which works for "good" maps. The author notes that the algebraic approach can be extended to the algebraic and complex cases. The text includes notation and definitions, such as the use of jet spaces, flags of bundles, and the total tangent bundle. It also mentions the use of sheaf theory concepts, although no results from sheaf theory are used. The author thanks several colleagues for their contributions to the work.This text is an introduction and overview of the study of singularities of differentiable maps, written by J. M. Boardman. It discusses the classification of singularities of smooth maps between manifolds, focusing on the singularity subsets and their properties. The author introduces the concept of jet spaces and uses them to construct subsets of the jet space that are independent of any specific map. These subsets induce subsets of the domain manifold, and the main theorem states that these subsets are submanifolds in the jet space, implying that for "sufficiently good" maps, the induced subsets are submanifolds of the domain manifold. The codimensions of these subsets are computed, and it is shown that they coincide with those obtained by a naive geometric approach. The infinite jet space and its associated total tangent bundle are used extensively. The text also discusses the inductive procedure for constructing the singularity sets and intrinsic derivatives, which works for "good" maps. The author notes that the algebraic approach can be extended to the algebraic and complex cases. The text includes notation and definitions, such as the use of jet spaces, flags of bundles, and the total tangent bundle. It also mentions the use of sheaf theory concepts, although no results from sheaf theory are used. The author thanks several colleagues for their contributions to the work.