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This thesis is devoted to the construction and investigation of some properties of pairs of linear operators (A, B) (representations) satisfying commutation relations AB=BF(A), where F denotes certain polynomial. Commutation relations of this kind are called "covariance type" and finding their representations is equivalent to solving operator equations. This kind of commutation relations and operator representations on finite-dimensional or infinite-dimensional linear spaces are important in Physics, Engineering and many areas of Mathematics. The most original part of the present thesis contains the construction of representations of commutation relations AB=BF(A) by linear integral operators, multiplication operators and weighted composition operators defined on Banach spaces of continuous functions, on L_p spaces and l_p spaces. Conditions on kernels of integral operators and functions defining multiplication and composition operators are derived for these operators to satisfy the covariance type commutation relations, both for the general polynomials F and for important specific choices of F. These conditions are used for construction of various concrete pairs of operators satisfying such commutation relations. Many essential algebraic properties of commutation relations are studied here as well.