This paper is a slightly reduced and concise version of a course given at Geonet2019 school on New Trends in Mathematical Methods for Physics at IMSP in Rep. of Benin, May 2019. The notes are intended to provide the reader with an introduction of a more recent procedure of quantization that is a generalization of the coherent states quantization procedure, highlighting the link between symplectic geometry and classical mechanics. The topic and the goal of the Geonet2019 school motivated the choice of the title of the course as New Trend in Quantization Methods. It is also a coincidence of a current research interest in integral quantization.