In this paper we presented some algorithms for fast QR matrix factorization with column pivoting (QRCP factorization). The QR factorization runs very fast on modern computers due to memory hierarchy and parallel execution on multiple CPU cores by using blocked algorithm. WY representation of the product of Householder reflectors is used for this purpose. The algorithm is implemented in LAPACK routine dgeqrf. The QRCP factorization is not much more complex than the QR factorization, compared by the number of arithmetic operations needed, but it is hard to preform into a blocked algorithm because we cannot immediately decide about the next \(b\) pivot columns. Partial solution is the blocked QRCP factorization which is implemented in LAPACK routine dgeqp3. It does not provide multiple speed-up because it updates only one row in each step. We showed some applications of the QRCP factorization, and described the application in finding lower rank approximations where the QRCP factorization often gives results comparable to optimal, but computationally more complex, singular value decomposition. We presented three other approaches to the QRCP factorization, that are much more time efficient than the classical QRCP factorization, and some variations of them. What makes them fast is the fact they do not require the column with the biggest norm in each step. The first algorithm is the QR factorization with controlled local pivoting, each uses predetermined worst allowed matrix condition number. We search for the best pivot candidates in a current block, until each of them causes low condition number. Then the current block is expanded by the new columns. The second one is communication avoiding algorithm, which divides matrix to column blocks on which the QRCP factorization will be done in parallel, and it finds the best choice for \(b\) pivot columns. The third algorithm is the randomized QRCP factorization which multiplies matrix A from the left by \(\Omega\), to decrease the number of rows. On that matrix we perform the classical QRCP factorization, which is used to detemine order of pivot columns in the initial matrix \(A\). If \(A\) is not thin enough, randomized sampling must be repeated which can be done by repeated random number generation, or by using, already existing, random sample. Two versions of the randomized QRCP factorization are implemented. We tested them and confirmed the expectation that their results will be comparable to the classical QRCP factorization by quality with significant speedup.