With the development of big data and cloud computing, the issue of secure search via the technique of searchable encryption has increasingly been the focus of the researchers in cryptography and network security all over the world. In the light of the new theories, new solutions and new techniques of searchable encryption, this paper presents a survey mainly from the following four aspects: the modes, the security, the expressiveness and the efficiency of secure searchable encryption. It discusses the new theories which are essential to secure search for ubiquitous network, including searchable encryption, attribute-based encryption, and applying these cryptographic mechanisms to obtain the generalized solutions to the theoretical problems of secure search in types of new emerging network services. Based on the aforementioned theoretical results, this paper studies the new approaches to construct practical secure search for these network services, comprising the light-weight public-key cryptographic algorithms, reducing the times of applying the light-weight public-key cryptographic algorithms in secure search, and exploiting any public-key cryptographic algorithm only once to obtain new approaches for secure search in the environment of resource-constrained network applications. We also focus on studying how to apply the new theories and approaches to solve the problems associated to secure search in different kinds of networks, including body area network, wireless vehicular ad hoc network, smart grid and so on. It is traditionally required to apply inefficient public-key cryptographic algorithms a number of times to construct secure search protocols. How to manipulate the public-key cryptographic algorithms and make them suitable to be used in resource-constrained networks becomes the key issue. Light-weighting public-key cryptographic algorithms is certainly a convincing way to address it. On the other hand, minimizing the number (once would be ideal) of applying the light-weighted public-key cryptographic algorithms guarantees more efficient and practical solutions and thus is the key problem to address the issue. Finally, we suggest several interesting open research issues and the trend in the future.