  A step forward is made in a long standing Lov\'{a}sz's problem regarding
hamiltonicity of vertex-transitive graphs by showing that every connected
vertex-transitive graph of order a product of two primes, other than the
Petersen graph, contains a Hamilton cycle. Essential tools used in the proof
range from classical results on existence of Hamilton cycles, such as
Chv\'atal's theorem and Jackson's theorem, to certain results on polynomial
representations of quadratic residues at primitive roots in finite fields.
