Contact geometry is a venerable subject that arose out of the study of Geometric Optics in the1800s. Though the years it has repeatedly cropped up in many areas of mathematics, but only in thepast 30 years or so has it received serious attention. Recently there has been great progress inunderstanding contact structures. Depending on ones perspective contact structures sometimes seemlike topological objects, sometimes geometric objects and sometimes dynamical objects. In this talkI will begin by discussing how contact structures arise out of natural problems and how they have deep connections with topology and dynamics. Then after surveying a few topics about contact structures in low dimensions I will define contact homology in certain situations. Contact homologyis a new invariant of contact structures (and/or certain submanifolds of them) that is similar, in spirit, to Gromov-Witten invariants of symplectic manifolds or Floer homology of Lagrangiansubmanifolds in symplectic manifolds. I then will proceed to discuss applications of contact homology, in particular, I will describe how it yields potentially new invariants of submanifolds of Euclidean space.