The idea of singularity is found in many parts of mathematics, capturing the idea of a position where some regular behavior breaks down. A standard situation is in linear algebra where for a linear transformation or matrix rank deficiency corresponds to singularity. This provides a basic model for other settings, especially for differentiable functions between Euclidean spaces. Results of Marston Morse (the Morse Lemma) and Hassler Whitney (stable singularities of mappings from the plane to the plane) led to pioneering work in differential topology by René Thom, whose interest in biological morphogenesis gave rise to Elementary Catastrophe Theory and a wide interest in mathematical models of singularity. The foundations of singularity theory were developed, in which the concepts of transversality and stratification played an important role. In the first part of the seminar I will outline some of this history and ideas from singularity theory. In robotics, kinematic mappings relate inputs, outputs and constraints. The impact of singularities on robotic control systems was recognised in the 1960s. Subsequent interest in the variety of ways that singular phenomena occur in robot kinematics has led to a large literature on the subject. In the second part of the seminar, I will discuss some of my research on kinematic singularities.