Resources for learning Markov chain and hidden Markov models I am looking for resources (tutorials, textbooks, webcast, etc) to learn about Markov Chain and HMMs. My background is as a biologist, and I'm currently involved in a bioinformatics-related project. 
Also, what are the necessary mathematical background I need to have a sufficient understanding of Markov models & HMMs?
I've been looking around using Google but, so far I have yet to find a good introductory tutorial. I'm sure somebody here knows better.
 A: It is quite surprising to see that none of the answers mention the Rabiner tutorial paper on HMMs.
While the practical implementation (the latter part of the paper) is focused on speech recognition, this paper is probably the most commonly cited one in the HMM literature, thanks to its clear and well-presented nature.
It starts by introducing markov chains and then moves on to HMMs.
A: For bioinformatics applications, the classic text on HMMs would be Durbin, Eddy, Krough & Michison, "Biological Sequence Analsysis - Probabilistic Models of Proteins and Nucleic Acids", Cambridge University Press, 1998, ISBN 0-521-62971-3.  It is technical, but very clear and I found it very useful.
For MCMC there is a recent (version of a) book by Robert and Casella, "Introducing Monte Carlo Methods with R", Springer, which looks good, but I haven't had a chance to read it yet (uses R for examples, which is a good way to learn, but I need to learn R first ;o)
A: Already nice suggestions, I would like to add the following articles that describe HMMs from perspective of application in biology by Sean Eddy. 


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*Hidden Markov Models 

*Profile hidden Markov models 

*What is a hidden Markov model?
A: I learned HMMs using the great book by Walter Zucchini and Iain L. MacDonald 
Hidden Markov Models for Time Series: An Introduction Using R
It's really good and features examples in R.
A: Take a look at the (HMM) Toolbox for Matlab by Kevin Murphy and also section Recommended reading on HMMs on this site.
You can also get Probabilistic modeling toolkit for Matlab/Octave with some examples of using Markov Chains and HMM.
You can also find lectures and labs on HMM, for example:


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*Labs

*Lecture1 and Lecture2
A: My 2 cents 
Beautifully explained and free.
Hidden Markov Models, Theory and Applications
University of Leeds Tutorial
A: Here are some notes by Ramon van Handel at Princeton: 

This course is an introduction to some of the basic mathematical,
  statistical and computational methods for hidden Markov models.

The first section includes a nice set of applications of HMMs in biology, finance,...
A: Here is a nice interactive introduction to Markov Chains http://setosa.io/ev/markov-chains/
A: Here are some tutorials (available as PDFs):


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*Dugad and Desai, A tutorial on hidden markov models

*Valeria De Fonzo1, Filippo Aluffi-Pentini2 and Valerio Parisi (2007). Hidden Markov Models in Bioinformatics. Current Bioinformatics, 2, 49-61.

*Smith, K. Hidden Markov Models in Bioinformatics with Application to Gene Finding in Human DNA
Also take a look at Bioconductor tutorials. 
I assume you want free resources; otherwise, Bioinformatics from Polanski and Kimmel (Springer, 2007) provides a nice overview (§2.8-2.9) and applications (Part II).
A: There is also a really good book by Oliver Cappe et. al: Inference in Hidden Markov Models.  However, it is fairly theoretical and very light on the applications.
There is another book with examples in R, but I couldn't stand it - Hidden Markov Models for Time Series.
P.s. The speech recognition community also has a ton of literature on this subject.
A: There are only 3 video i have found very useful for understanding maths behind hidden markov models-  
https://www.youtube.com/watch?v=E3qrns5f3Fw
https://www.youtube.com/watch?v=cjlhpaDXihE
https://www.youtube.com/watch?v=5sGEF-e82yY
These are really good and taught by one the best Indian professors from IIT krg.
A: This playlist is a great explanation and is based on the paper by Rabiner mentioned in the answer above. -
https://www.youtube.com/watch?v=J_y5hx_ySCg&list=PLix7MmR3doRo3NGNzrq48FItR3TDyuLCo
This above playlist is a 12 series lecture which begins with explanation of Markov Chains/ observable Markov Models and then moves on to HMMs
