In this paper we present the results of a survey of research studies on the learning of multivariable calculus in a wide geographic spectrum. The goal of this study is to describe what research results tell us about students' learning of calculus of two-variable functions, and how they inform its teaching. In spite of the diversity of cultures and theoretical approaches, results obtained are coherent and similar in terms of students' learning, and of teaching strategies designed to help students understand two-variable calculus deeply. Results show the need to introduce students to the geometry of three-dimensional space and vectors, the importance of the use of graphics and geometrical representations, and of thorough work on functions before introducing other calculus topics. The reviewed research deals in different ways with the idea of generalization concerning how students' knowledge of one-variable calculus influences their understanding of multivariable calculus. Some research-based teaching strategies that have been experimentally tested with good results are included. Results discussed include the following: basic aspects of functions of two variables, limits, differential calculus, and integral calculus. The survey shows that there is still a need for research using different theoretical perspectives, on the transition from one-variable calculus to two-variable and multivariable calculus.