Matching problems with group-fairness constraints and diversity constraints have numerous applications such as in allocation problems, committee selection, school choice, etc. Moreover, online matching problems have lots of applications in ad allocations and other e-commerce problems like product recommendation in digital marketing. We study two problems involving assigning {\em items} to {\em platforms}, where items belong to various {\em groups} depending on their attributes; the set of items are available offline and the platforms arrive online. In the first problem, we study online matchings with {\em proportional fairness constraints}. Here, each platform on arrival should either be assigned a set of items in which the fraction of items from each group is within specified bounds or be assigned no items; the goal is to assign items to platforms in order to maximize the number of items assigned to platforms. In the second problem, we study online matchings with {\em diversity constraints}, i.e. for each platform, absolute lower bounds are specified for each group. Each platform on arrival should either be assigned a set of items that satisfy these bounds or be assigned no items; the goal is to maximize the set of platforms that get matched. We study approximation algorithms and hardness results for these problems. The technical core of our proofs is a new connection between these problems and the problem of matchings in hypergraphs. Our experimental evaluation shows the performance of our algorithms on real-world and synthetic datasets exceeds our theoretical guarantees.