In the thesis products and sums of nilpotent operators, square-zero operators and quasi-nilpotent operators are studied. We consider square matrices over arbitrary field, linear bounded operators on an infinite-dimensional, separable, complex Hilbert space and in some cases linear transformations on an infinite-dimensional vector space. In the first part of the thesis we present products of two or more nilpotent operators. Every singular matrix that is not $2 \times 2$ non zero nilpotent can be written as a product of two nilpotent matrices. A necessary and sufficient condition for an operator to be a product of two nilpotent operators on an infinite-dimensional Hilbert space is that its kernel and its co-kernel are infinite-dimensional. Next, we deal with products of square-zero operators. We show that a matrix $T$ is a product of two square-zero matrices if and only if the codimension of $\mathrm{ker}T \cap \mathrm{im} T$ in $\mathrm{ker}$ is greater than or equal to $\mathrm{rank} T$. In the case of operators on a Hilbert space we also find a necessary condition and a sufficient condition. Besides, we show a characterization of products of two square-zero linear transformations on an infinite-dimensional vector space. For the products of two quasi-nilpotent operators a necessary and sufficient condition is that the operator is not semi-Fredholm. We also study products of commuting nilpotent and square-zero operators. We characterize products of two commuting nilpotent matrices and linear transformations. In the case of square-zero operators we characterize products of two or more commuting square-zero matrices and operators. We also consider a question which operators can be expressed as a sum of two square-zero operators. A matrix $T$ is a sum of two square-zero matrices if and only if it is similar to $-T$. We characterize those invertible ( resp. normal) operators on a Hilbert space which are sums of two square-zero operators. We also study sums of commuting square-zero matrices and linear transformations. For sums of quasi-nilpotent operators it is shown that an operator is a sum of two quasi-nilpotent operators if and only if it is a commutator. In addition, every operator is a sum of three quasi-nilpotent operators.