The paper is devoted to the development of the mathematical model of topological insulators (TIs), based on index theory. TIs present solid state structures of independent electrons with Fermi level, locating in a mobility gap and topologically non-trivial Fermi projection, which cannot be deformed similarly to a normal insulator. The Fermi projections have non-trivial topological invariants (given, for example, in terms of winding numbers and Chern numbers). At the boundaries of a topological material, there are delocalized surface modes, that make the insulator conducting. First, a short description of the physics and the underlying quantum mechanical models of the TIs are presented with a subsequent overview of recent mathematical results on them. Into the framework of tight-binding approximation, an important physical phenomenon linked to random potentials and random quantum Hamiltonians in general is discussed, namely the Anderson localization. A projection identified with a vector bundle over the torus and topological invariant is the Chern number, associated to this vector bundle. The winding number is equal to the Chern number, defining the bulk-boundary correspondence (BBC). This forms a \(K\)-theoretical mathematical description, applied to disordered systems. From the mathematical viewpoint, the aim is first to distinguish and classify Fermi projections with time-reversal symmetry (TRS) that is achieved by using the \(K\)-theory with symmetries, which follows the \(KR\)-theory. The second aim is then to calculate the invariants and to analyze the physical effects that go along with these invariants. This is achieved by considering particle-hole symmetry (PHS) and combining the TRS and PHS, leading to a chiral symmetry (CHS). The fermionic one-particle bulk Hamiltonians are studied by using associated observable algebras. Then \(K\)-theory is attracted to analyze homotopy classes of projections in the algebras and is used to distinguish topological phases. One group of \(K\)-theory is used to classify \(D\)-dimensional fermionic systems with CHS. Two complex \(K\)-groups allow a classification of the Fermi projections of gapped covariant systems without further symmetry and an approximate CHS. The paper calculates these groups and discusses their structures. The non-commutative analysis is used for the definition of topological invariants and their main properties. It is shown that the strong invariants and the associated indices can be used as a phase label in the localization regime. Then, another key feature of topological insulators is treated which shows that the invariants are responsible for different effects linked to defects. Due to this, BBC is considered based on the half-space Hamiltonian. Two examples (quantization of boundary currents and anomalous surface quantum Hall effect) are considered as consequences of the general theorem on the duality of pairing, associated with the exact sequences of algebras. Reversal symmetry and PHS are considered; both of which require a real structure, allowing to define the complex conjugate of an operator. Then, the bound states attached to particular types of point defects in TIs are considered. In particular, the spectral flow in a 2D system with odd TRS (i.e., in a quantum spin Hall system) is discussed. Finally, the use of spin Chern numbers is discussed, being the topological invariants for distinguishing phases of systems, having exact symmetries (like TRS and PHS).