  The following paper is inspired by Efimov's problem - an undecided problem of
whether there exists an infinite compact topological space that does not
contain neither non-trivial convergent sequences nor a copy of $\beta\omega$.
After introducing basic topological concepts, we present several classes of
topological spaces in which such sequences can certainly be found, namely
ordered, scattered, metrisable spaces and Valdivia compacta. We show that some
cardinal coefficients set limits on the smallest cardinality of the base and
the smallest cardinality of a neighbourhood base, under which the existence of
convergent sequences can be ensured. In the final part of the paper we define
the space $\beta\omega$ and show its selected properties. In particular, we
prove that there are indeed no non-trivial convergent sequences in
$\beta\omega$. Whereas the statements of these theorems are commonly known, the
proofs are notoriously difficult to find. In this paper we intend to fill that
gap.
