This work is focused on the distributed system, i.e. Multi-agent Systems (MAS), with application in environmental monitoring and learning. The specific task is to develop algorithms, i.e. Gaussian Process (GP), that are robust, accurate and fully-distributed to learn the unknown spatial environmental field. The two main problems are (1). how to optimize GP hyperparameters in fully-distributed manner, and (2). how to aggregate predictions from agents. The state-of-the-art solution for distributed GP hyperparameter optimization problem is proximal alternated direction method of multipliers (pxADMM) algorithm, which requires a center station in MAS. Based on pxADMM, two fully-distributed pxADMM algorithms are proposed such that the center station is no longer needed. Asynchronous behavior is also introduced into the proposed algorithms, so that they can deal with heterogeneous processing time of agents. Simulations are carried out on both artificial and real datasets. Results show that the proposed methods all achieve stable convergence. The aggregation methods can be classified based on whether the local datasets are assumed to be independent or not. Under independent assumption, PoE and BCM families of methods can be distributed by applying discrete time consensus filter (DTCF). In this project, primal-dual method of multiplier (PDMM) is proposed to replace DTCF so that the aggregation converges faster. Without independent assumption, the Nested Pointwise Aggregation of Experts (NPAE) considers the cross-correlation among local datasets to achieve consistent aggregation. The current NPAE-JOR algorithm distributes NPAE in complete graph, where flooding variables across network is required before aggregation. In this project, CON-NPAE is proposed to extend NPAE to fully-distributed version in connected graph, where flooding is not required. Simulations on artificial and real datasets are performed. Results show that the proposed PDMM based algorithm reduces the iterations needed for fully-distributed PoE and BCM families of methods. The CON-NPAE is fully-distributed and makes better aggregations than independence assumption based methods in networks with high connectivity. The connection between its performance and MAS structure requires further study. In conclusion, fully-distributed and asynchronous algorithms are proposed for GP hyperparameter optimization based on pxADMM. The fully-distributed PoE and BCM methods are accelerated by applying PDMM. The CON-NPAE is proposed to make NPAE fully-distributed. In future work, the theoretical convergence of fully-distributed pxADMM should be researched. For GP aggregation, the effect of network structure on the performance of CON-NPAE can be studied. Also, inducing points method is a possible solution to alleviate the flooding overhead of distributed NPAE.