An operad O can be thought of as a sequence of spaces O(n) which encode “n-ary operations” satisfying certain compatibility axioms for compositions, units, and actions by the symmetric group. Different operads encode different properties that n-ary operations might possess: for instance, the “commutative operad” Comm encodes what it means for an n-ary operation to be (“on the nose”) commutative, while the associative operad Assoc encodes (“on the nose”) associativity. Motivated by the homotopy-associative property of loop concatenation in loop spaces, we’ll introduce A_infinity operads and algebras over them. We’ll finish the talk with the little n-cubes operad and algebras over them (E_n / E_infinity spaces), which possess a “higher homotopy” analogue of commutativity.