The construction of multilinear operations, called braces, on the Hochschild complex of an associative algebra leads to the definition of a brace algebra structure on a graded vector space. We will develop the concept of a symmetric brace algebra in which the brace operations possess the property of graded symmetry. Multilinear operations on the space of anti-symmetric maps on a graded vector space, e.g. the Chevalley-Eilenberg complex of a Lie algebra, provides a motivating example of this concept. The relation of brace and symmetric brace algebras to pre-Lie algebras will also be discussed.