This book presents an introduction to the local differential geometry of curves and surfaces in Euclidean space. All standard parts of the theory (plane curves, curves in three-dimensional space studied by Frenet frame methods and surfaces in three-dimensional space) are included but there are also some additional topics. Both the style of the book and its content are inspired by ideas and the work of R. Thom and V. I. Arnold on singularity theory. Several topics related to singularities such as umbilics, cusps, probes and ridges and ribs for surfaces are presented in details. Topics discussed in the book include moreover plane kinematics, curves on the unit sphere, multi-linear forms and their applications in geometry, probes and contacts. In surface theory, ridge and rib points as well as umbilics are studied very intensively. The discussion of parabolic and subparabolic lines on a regular surface is interesting. Some concepts and results of V. I. Arnol'd regarding curves on S2 and its extension to curves and surfaces on S3 can be found in the last chapter. There are many historical notes and comments throughout the book as well as some curiosities. It is a very good and interesting introduction to differential geometry of curves and surfaces, which can be recommended to anybody interested in the subject.