In mathematics there are many interesting number sequences, such as odd and even numbers, prime numbers, square and cube numbers, Fibonacci numbers etc. This thesis studies certain numbers which are specific for the area of combinatorics, in particular binomial coefficients, \(q\)-binomial coefficients (Gaussian coefficients), Stirling numbers and \(q\)-Stirling numbers, along with brief mention of Bell numbers. The binomial coefficient \(\binom {n}{k}\) is the number of \(k\)-element subsets of a set of \(n\) elements. The \(q\)-analogue of the binomial coefficient, also called a Gaussian coefficient \({n \brack b}_q\), counts the number of \(k\)-dimensional subspaces of an \(n\)-dimensional vector space over the finite field of order \(q\). The Stirling number of the second kind \({n \brace k}\) counts \(k\)-partitions of an \(n\)-element set, in other words the ways to divide a set of \(n\) elements into \(k\) nonempty subsets. Bell number \(B_n\) counts the number of all partitions of an \(n\)-element set. The \(q\)-Stirling number \({n \brace k}_q\) is defined as a weight of the set of all \(k\)-partitions of an \(n\)-element set. In the thesis we give various identities and relations valid for the aforementioned combinatorial numbers. We emphasise similarities and differences between numbers enumerating finite sets and their analogues over finite fields.