Spectral asymptotics of a tensor product of compact operators in Hilbert space with known marginal asymptotics is studied. Methods of A. Karol', A. Nazarov and Ya. Nikitin (Trans. AMS, 2008) are generalized for operators with almost regular marginal asymptotics. In many (but not all) cases it is shown, that tensor product has almost regular asymptotics as well. Obtained results are then applied to the theory of small ball probabilities of Gaussian random fields. §1 IntroductionWe consider compact nonnegative self-adjoint operators T = T * 0 in a Hilbert space H and T in a Hilbert space H. We denote by λ n = λ n (T ) the eigenvalues of the operator T arranged in a nondecreasing order and repeated according to their multiplicity. We also consider their counting functionSimilarly we define λ n and N (t) for T .Having known asymptotics for N (t, T ) and N (t, T ) as t → 0, we aim to determine the asymptotics for N (t, T ⊗ T ). Obtained results are easily generalized to the case of a tensor product of multiple operators.Known applications of such results could be found in problems concerning asymptotics of random values and vectors quantization (see e.g. [1, 2]), average complexity of linear problems, i.e. problems of approximation of a continuous linear operator (see e.g. [3]), and also in the developing theory of small deviations of random processes in L 2 -norm (see e.g. [4,5]).Abstract methods of spectral asymptotics analysis for tensor products, generalized in this paper, were developed in [4] and [5]. In [4] the case is considered, in which the eigenvalues of the operators-multipliers have the so-called regular asymptotic behavior: λ n ∼ ψ(n) n p , n → ∞,