This is a good question with many different answers.

For an overview of MOL, see

J.R. Cash, Y. Psihoyios, The MOL solution of time dependent partial differential equations, Computers Math. Applic. 31 (11), 1996, 60-78:

http://ac.els-cdn.com/0898122196000636/1-s2.0-0898122196000636-main.pdf?_tid=012520fa-70e5-11e4-80d9-00000aacb35d&acdnat=1416509139_c2fbf876f5720584a50d194528936898

See (1)-(4), p. 69 for the MOL model.    There are 23 instances in this article where MOL is discussed.   An overview of MOL software is given on p. 70.

The numerical method of lines via Mathematica is given in great detail in

http://reference.wolfram.com/language/tutorial/NDSolveMethodOfLines.html

A combination of MOL and the Runge-Kutta method is considered in an interesting way in

K.J. Wojciechowski, Analysis and Numercial Solution of Nonlinear Volterra Partial Integrodifferential Equations Modelling Swelling Porous Materials, Ph.D. thesis, University of Colorado Denver, 2011:

http://www.ucdenver.edu/academics/colleges/CLAS/Departments/math/students/alumni/Documents/Student%20Theses/Wojciechowski_PhDThesis.pdf

See Section 5.1.2, starting on page 108.   See, for instance, Example 3, p. 112.   This example reveals that the MOL approach with an RK4 solver gives a competitive performance for smooth kernels.

The method of lines is used for sloving PDEs. Most often it refers to the construction or analysis of numerical methods for partial differential equations that proceeds by first discretizing the spatial derivatives only and leaving the time variable continuous. This leads to a system of ordinary differential equations to which a numerical method for initial value ordinary equations can be applied

There are many references you may refere to:

Causley, M., Christlieb, A., Ong, B., & Van Groningen, L. (2014). Method of lines transpose: An implicit solution to the wave equation. Mathematics of Computation.

Dereli, Y., & Schaback, R. (2013). The meshless kernel-based method of lines for solving the equal width equation. Applied Mathematics and Computation, 219(10), 5224-5232.

Northrop, P. W., Ramachandran, P. A., Schiesser, W. E., & Subramanian, V. R. (2013). A robust false transient method of lines for elliptic partial differential equations. Chemical Engineering Science, 90, 32-39.

Furzeland, R. M., Verwer, J. G., & Zegeling, P. A. (1990). A numerical study of three moving-grid methods for one-dimensional partial differential equations which are based on the method of lines. Journal of Computational Physics, 89(2), 349-388.

Reddy, S. C., & Trefethen, L. N. (1992). Stability of the method of lines. Numerische Mathematik, 62(1), 235-267.

Rektorys, K. (1982). The method of discretization in time and partial differential equations. Equadiff 5, 293-296.

http://people.maths.ox.ac.uk/trefethen/publication/PDF/1992_50.pdf

http://oai.cwi.nl/oai/asset/1640/1640A.pdf

http://dml.cz/bitstream/handle/10338.dmlcz/702309/Equadiff_05-1982-1_64.pdf

http://num.math.uni-goettingen.de/bibi/preprintserie/preprints/schaback/TMKBMoLfStEWE_V2.pdf

http://www.maple.eece.wustl.edu/pubs/45_CES_False_Transient_PDE.pdf

http://mathgeek.us/research/papers/1DWavePaper.pdf

See http://abs-5.me.washington.edu/pub/370282.pdf

You can See this two links :

http://www.pdecomp.net/Scholarpedia/MOLfinal.pdf

http://people.math.gatech.edu/~meyer/MOL-notes/Chap1.pdf

Method of lines is simply used to convert PDE into ODE .

hello

you can see my papers in the MOL.

It is a semi-analytical semi-numerical method that can analyze multilayered structures.

Best Wishes