How are they defined? How many people belong to them?
The terms Renewal and Renewalists are broad enough to encompass these three worldwide movements:
First Wave: Pentecostal (early 20th century →)
Second Wave: Charismatic (early 1960s →)
Third Wave: Neo-Charismatic / Independent Charismatic (late 1970s and 1980s→)
Fourth Wave: Is another one coming, or is it just a resurgence and advance of the first three?
Please note that the arrows mean that the movements are still going on.
Also, the Jesus Revolution was contemporaneous with the Charismatic Renewal from the late 1960s and throughout the 1970s. The Jesus Movement was for the rootless, drifting young people, while the Charismatic Renewal was for the established, church-going, older generation.
Numerous subcategories of the first three waves are possible. They all have different theological emphases, but they share in common the fullness and ongoing operation of the Spirit with power, including signs and wonders and other gifts. They all share in common a robust and biblical pneumatology (doctrine of the Spirit). They believe in the baptism with the Spirit or the “empowerment with the Spirit.” This experience is distinct from the born again experience or regeneration, which happens once. The baptism or empowerment of the Spirit can happen many times.
Pentecostals believe that the initial baptism with the Spirit, subsequent to regeneration, must be signified by speaking in “tongues” or Spirit-inspired languages. The other movements believe that the baptism with the Spirit is commonly or frequently accompanied with this gift, but is not a requirement.
Those three massive movements are growing faster than any other sector of the global church. In 2000 the estimated total of all three waves was over 523 million members, and it is predicted they will reach over 811 million by 2025. (Source: International Dictionary of Pentecostal and Charismatic Movements, rev. and expanded edition).
However, we are now nearing 2025, and a newer 2020 report says:
Globally, the movement includes 644 million people, about 26 percent of all Christians, according to a new report from the Center for the Study of Global Christianity. (Source)
Source: Center for Global Christianity. HT: Christianity Today
Here is how the website Center for Global Christianity defines Pentecostal and Charismatic and Independent (Neo-Charismatic) Christians.
The case for the Pentecostal and charismatic renewal as a single interconnected phenomenon can best be made by considering a “family resemblance” among the various kinds of movements that claim to be either Pentecostal or charismatic. For the purpose of understanding the diverse global phenomenon of Pentecostalism, it is useful to divide the movement into three kinds or types. First are denominational Pentecostals, organized into denominations in the early part of the twentieth century and defined as Christians who are members of the explicitly Pentecostal denominations whose major characteristic is a new experience of the energizing ministry of the Holy Spirit that most other Christians have considered to be highly unusual. This is interpreted as a rediscovery of the spiritual gifts of New Testament times and their restoration to ordinary Christian life and ministry.
Second are Charismatics, individuals in the mainline denominations (primarily after the mid-twentieth century), defined as Christians affiliated to non-Pentecostal denominations (Anglican, Protestant, Catholic, Orthodox) who receive the experiences above in what has been termed the charismatic movement.
Third are Independent Charismatics, those who broke off of denominational Pentecostalism or mainline denominations to form their own networks. While the classification and chronology of the first two types is straightforward, there are thousands of churches and movements that “resemble” the first two types but do not fit their definitions. These constitute a third type and often predate the first two types.
Pentecostals and charismatics are located globally by using a taxonomy of the world’s denominations. First, each major tradition of Christianity (Roman Catholic, Orthodox, Independent, Protestant) is sub-divided into minor traditions (e.g., Protestants as Lutherans, Baptists, Presbyterians, etc.). Pentecostals and charismatics appear within denominations in three ways. First, among Protestants, are classical Pentecostal denominations; second, Pentecostals outside of the Western world; third, charismatic individuals within non-Pentecostal denominations. … . (Source)
The description is of course excellent, but the three main movements or branches can be labeled in a bundle: Renewalist. So Pentecostals and Charismatics and Independent Charismatics (or Neo-Charismatics) fit under the larger term Renewal Movements.
The church would be impoverished without them.