In this thesis we will define a new shape category \(Sh^{\circledast}\) of topological spaces called the finite coarse shape category. Category \(Sh^{\circledast}\) will have the same class of objects as the categories of shape and coarse shape, but morphisms will be different. We will contruct the finite coarse shape category by using polyhedral expansions and inverse systems theory, i.e., using an external approach. Two appropriate faithful functors will be defined, one of them from the shape category \(Sh\) to the finite coarse shape category \(Sh^{\circledast}\) and the other one from the finite coarse shape category \(Sh^{\circledast}\) to the coarse shape category \(Sh^*\). We will prove by the examples that those functors are not full, i.e., that the finite coarse shape category is a proper subcategory of the coarse shape category and that it contains the shape category as its proper subcategory. The coarse shape category of metric compacta will also be given an intrinsic approach. To achieve that, we will restrict the class of objects to the set of all closed subsets of the Hilbert cube \(Q\) and replace the inverse systems theory by the theory of \(\epsilon\)-continuous functions. Therefore we shall give a detailed overview of the \(\epsilon\)-continuity theory by defining the main notions and proving the most important properties of the \(\epsilon\)-continuous functions and \(\epsilon\)-homotopy. We will generalise Borsuk's fundamental and approximative sequences and Sanjurjo's proximate sequences by defining so called \(\circledast\)-fundamental, \(\circledast\)-approximative and \(\circledast\)-proximate sequences, respectively. On the sets of the \(\circledast\)-fundamental, \(\circledast\)-approximative and \(\circledast\)-proximate sequences between any two closed subsets of the Hilbert cube \(Q\) we will define an appropriate equivalence relations, classes of which will be morphisms of the new categories \(Sh^{\circledast}_f\), \(Sh^{\circledast}_a\) and \(InSh^{\circledast}\), respectively. Category \(InSh^{\circledast}\) will be called the intrinsic finite coarse shape category}. Futhermore, three new functors will be defined, one of them between the categories \(Sh^{\circledast}|_Q\) (the restriction of the \(Sh^{\circledast}\) category to the set of all closed subsets of \(Q\)) and \(Sh^{\circledast}_f\), the second one between \(Sh^{\circledast}_f\) and \(Sh^{\circledast}_a\) and the third one between the categories \(Sh^{\circledast}_a\) and \(InSh^{\circledast}\). We will prove that each of these three functors is an isomorphism, which means that category \(InSh^{\circledast}\) is an intrinsic reinterpretation of the finite coarse shape category \(Sh^{\circledast}|_Q\) of all closed subsets of \(Q\). We will also define a faithful functor from \(InSh\) to \(InSh^{\circledast}\) which is an identity on the set of objects and which associates each intrinsic shape morphism with the induced intrinsic finite coarse shape morphism. Hence, Sanjurjo's intrinsic shape category \(InSh\) may be considered as a proper subcategory of the constructed intrinsic finite coarse shape category \(InSh^{\circledast}\). In the last section we will prove that the intrinsic finite coarse shape does not depend on the embedding of a compact metric space into the Hilbert cube \(Q\), i.e., that every two embeddings of any compact metric space have the same intrinsic finite coarse shape. That will allow us to extend the classification by the intrinsic finite coarse shape to the whole class \(\mathcal{MC}pt\) of metric compacta. Finally, we will prove that two compact metric spaces \(M\) and \(M'\) have the same finite coarse shape if and only if \(M\) and \(M'\) have the same intrinsic finite coarse shape.