We start by introducing general determinantal point processes in one dimension and their relation to random matrices following Borodin. Several examples with increasing level of complexity will be discussed as the classical Gaussian Unitary Ensemble, products of several independent and uncorrelated Gaussian random matrices and the effect of introducing correlations. We will then display the corresponding double contour integral representation of the respective kernels and discuss the issue of universality in the limit of large matrix size. This is based on several joint works with Eugene Strahov as well as a work including also Tomasz Checinski and Dang-Zheng Liu.