We consider a strict factor model with high-dimensional observations and a relatively small number of factors. Firstly, I will report some new results on the problem of determination of the number of factors including a new method we devised recently. Next, we consider the goodness-of-fit test and derive new asymptotic distribution for the LR statistic. Our approach is entirely based on recent advancement in the theory of random matrices, in particular those related to the eigenvalues distribution of a large and spiked sample covariance matrices. Further questions on these problems will also be discussed.