In this thesis we described some basic terms, theorems and results about polynomial matrices. First of all, we had to define some terms which we had already known like polynomials itself, matrices and some basic operations on polynomials and matrices and after that, we gave the definition of the polynomial matrix. We described the sum of two polynomial matrices, the product of two polynomial matrices and the product of a polynomial matrix and a scalar. After those basic definitions we proved some important theorems about quotients and remainders and also gave some examples to confirm them. There is also an important theorem called Bézout’s theorem which we had generalized. We described linear independence of polynomial vectors and basis for them and also defined the rank of a polynomial matrix. We have noticed that those definitions are quite similar to those of ’classic’ matrices. We showed an example where were given the polynomial vectors which were linearly independent and in the other example they were linearly dependent. We also gave the definition of Smith canonical form of polynomial matrices and the definition of invariant polynomials which we had also used in other chapters. We gave definitions of equivalent matrices and explained the connection between similarity of classical matrices and their polynomial matrices of the first degree. Approaching to the end of the thesis, we defined zeros and elementary divisors of polynomial matrices and we saw that all these definitions are connected with the invariant polynomials. In the end, we defined Frobenius canonical form of a polynomial matrix and also saw some links between a polynomial matrix and its Frobenius canonical form.