Abstract: We shall explore the historical connection between Symmetry and Groups and also try to address current research questions concerning symmetry groups. In 1829-30, a nineteen-year-old mathematician, Évariste Galois made an important breakthrough. Mathematicians for centuries had been wrestling with the problem solving equations of degrees higher than 2. The quest was to find a `nice formula’ expressing the roots of the equations in terms of the coefficients. In the 17th century, Italian mathematicians had shown that there are such formulae for the cubic and the quartic equation. Abel had in 1824 shown that such a general formula was not possible for a quintic. Galois, however saw the shapes hidden in the roots of equations. This enabled him to realise that the symmetries associated with certain quintic equations were very different to those of the quadratic, cubic or quartic and lead him to `invent’ groups and use their properties to show the impossibility of a general formula for roots of polynomial equations of degree 5 or higher. The lecture will also dwell on `indivisible symmetry shapes’ and connect them to `finite simple groups’. Some glimpses of the large research project of the 20th century in classifying the finite simple groups will be given and also of current research that falls broadly in this area.