This course provides a basic introduction to conformal symmetry and supersymmetry, both of which play an important role in modern physical theories. Conformal transformations in arbitrary dimensions are characterised by leaving the space-time metric invariant up to a (local) scaling factor. They appear in a vast spectrum of physical applications, ranging from condensed matter to string theory and modern particle physics. Supersymmetry relates bosonic and fermionic fields to each other. While not yet directly observed at the energies of current collider experiments like the LHC, supersymmetry plays an important role in extending the standard model and in approaching important fundamental questions of modern high energy physics. This course starts with a mathematical description of both symmetries using a group theoretic language and explores their applications to field theories in various dimensions.