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Associative and unital algebras are well-known objects. In this talk, we are going to investigate their categorical duals called ''coalgebras.'' To motivate their use, we will introduce them purely computationally with several examples. After a brief introduction to simplicial and cyclic objects, and the two-sided bar complex of algebras and coalgebras, we will demonstrate a few cohomological computations. We will finish the talk by introducing algebra/coalgebra duality pairing on the level of simplicial/cyclic objects and bar complexes, and then investigate the use of such pairings in calculations of cohomology. (Parts of this talk are from our joint work with Serkan Sütlü from Işık University.)