Firstly is a tendency often referred to as the New Apostolic Reformation. Their focus is on a restoration of Ephesians 4 ministries (Apostles, Prophets, Evangelists, Pastors & Teachers), especially those of Apostles and Prophets. With these ministries is the expectation of signs and wonders, and an openness to ecstatic spiritual practice that some might find unusual. 

Theology would be broadly (but not exclusively) Arminian Evangelical with forays into territory more associated with the older Word of Faith movement, such as hyper-kenoticism (the belief that Christ's ministered entirely as an anointed human). Although many leaders are male, female leaders are clearly visible, and husband wife teams are common. I would trace much of the movements diversity to the coming together of different leaders and networks at the time of the 'Father's' or 'Toronto Blessing', and with this comes an element of new mysticism. 

Names associated with the New Apostolic are Peter Wagner who writes about it here, and Bethel Church headed by Bill Johnson who spoke at the New Wine Leaders Conference in 2007.

For many Calvinism and spiritual gifts seem like an unlikely combination. And yet in the United Kingdom the largest and most stable of the New Church movements is New Frontiers which combines Reformed theology with Charismatic practice and worship, and a leadership structure based on Ephesians 4. Further afield not all Reformed Charismatics are so explicit in their language of Apostles and Prophets, and some such as Mark Driscoll have expressed that they are 'Charismatic with a seatbelt'. 

Theology is clearly Reformed (although mostly baptist), with elements of Luther's thought as well as Calvin's found in parts. In general leadership is male with complimentarianism guiding different gender roles in ministry and marriage. With Reformed theology comes a certain culture of 'soundness' that takes exposition and systematic theology seriously, but can create a protective bubble that shields Reformed Charismatics from some of the excesses of the Charismatic movement.

Names associated with Reformed Charismaticism are Wayne Grudem whose Systematic Theology is a significant work, Terry Virgo and others from New Frontiers such as Adrian Warnock who has written on the movement here. John Piper is warmly supportive, and the perspective is gaining some traction amongst more Reformed Anglicans.

It is estimated that 160,000,000 Roman Catholics have been touched by Charismatic Renewal. Outside of the Roman Catholic church their are many others who hold a sacramental theology and the threefold orders of Bishop, Priest and Deacon who also identify with 'Renewal'. Catholic order acknowledges the ongoing apostolic ministry in Bishops, historically accepts the more diaconal ministries of prophet and evangelist, and sees the Pastor-Teacher role in a representative priesthood,

Charismatic Catholics have no particular theological emphasis within Catholic theology. Acceptance of the mystical and supernatural is widespread within Catholic theology and history with a natural confidence with Augustine and Patristic writers. Theological engagement is significant as evidenced by a Jesuit quoting Aquinas in this piece. Mysticism is embraced with Medjugorje being a well known place of pilgrimage. Views on leadership are influenced by understandings of sacramental ordination and if members of both sexes can represent Christ in the priesthood. As an Anglican Catholic I have experienced a warm ecumenicism from Roman Catholic Charismatics combined with theological orthodoxy.

Historically Cardinal Suenens wrote extensively on Renewal and Catholic faith, including a work with retired Anglo-Catholic Archbishop Michael Ramsey. Worship leader Matt Maher is Roman Catholic and came to faith through Catholic Charismatic Renewal. Charles Whitehead is known ecumenically, speaking at Anglo-Catholic Charismatic gatherings, and outlines his understanding of the renewal here.

These streams within the Charismatic movement have very different emphasis, and their differences demonstrate how difficult it is to criticise or embrace the movement as a whole. All three embrace a deeper experience of the Spirit but there are other areas of overlap too: The Reformed and the New Apostolic share a reshaping of Apostolic ministry, Reformed and Catholics share an Augustinian theological engagement, and the New Apostolic and Catholics meet in a renewed mysticism. All three, in various forms have been and are influencing Anglicanism and the Church of England.

And for the record. I started in No.2, passed through No.1 and ended up in No.3.