In this thesis we present the problem of smoothing on general state space hidden Markov models. Smoothing and other related problems with sequential structure have been solved by using the Kalman filter and the extended Kalman filter, which have strong assumptions on normality and linearity of models. Sequential Monte Carlo methods were developed with the goal of not needing such strong model assumptions. We present and motivate the development of an algorithm called sequential importance sampling with resampling, which serves as a prototype for all sequential Monte Carlo methods. The first chapter motivates exploration of hidden Markov models with examples, and tries to argue that it is a very versatile class of models. We also give a theoretical background on general state space Markov chains and formally define hidden Markov processes and the smoothing problem. Then in the second chapter, we present importance sampling and some of its extensions, with the goal of applying it on problems with a sequential structure, as in the case of smoothing in hidden Markov models. The extensions we present are self-normalizing importance sampling and importance sampling with resampling. For proving asymptotic normality of estimators that those algorithms give, we show generalizations of asymptotic normality for triangular arrays and tools which we use for the analysis of conditionally independent triangular arrays. In the third chapter we present sequential importance sampling and describe several possible ways of implementing instrumental kernels. That algorithm is a generalization of importance sampling which can be used for smoothing in hidden Markov models. We describe how weight degeneracy occurs in sequential importance sampling and present reasons why resampling is a necessary addition to sequential importance sampling. At the end, we present a proof of asymptotic normality of sequential sampling with resampling.