The development of massive open online courses (MOOCs) has launched an era of large-scale interactive participation in education. While massive open enrolment and the advances of learning technology are creating exciting potentials for lifelong learning in formal and informal ways, the implementation of efficient and effective assessment is still problematic. To ensure that genuine learning occurs, both assessments "for" learning (formative assessments), which evaluate students' current progress, and assessments "of" learning (summative assessments), which record students' cumulative progress, are needed. Providers' more recent shift towards the granting of certificates and digital badges for course accomplishments also indicates the need for proper, secure and accurate assessment results to ensure accountability. This article examines possible assessment approaches that fit open online education from formative and summative assessment perspectives. The authors discuss the importance of, and challenges to, implementing assessments of MOOC learners' progress for both purposes. Various formative and summative assessment approaches are then identified. The authors examine and analyse their respective advantages and disadvantages. They conclude that peer assessment is quite possibly the only universally applicable approach in massive open online education. They discuss the promises, practical and technical challenges, current developments in and recommendations for implementing peer assessment. They also suggest some possible future research directions.