In this survey talk I will present how the concepts coming from algebraic geometry find their relaxation in terms of topological spaces with toric action. In toric topology we put an accent on studying homotopy theoretical properties of these spaces which allow us to further widen a family of spaces we study to polyhedral products. In this way we are not limited only to toric actions but can consider an arbitrary group action. The beauty of the subject comes from its interplay with combinatorics, commutative algebra, and symplectic and algebraic geometry.