In this paper we would like to present a concept of conditional martingales given $\sigma$-field $\mathcal F$, which is a natural generalization of martingales. Based on $\mathcal F$-independence, we are able to show that there exist conditional martingales, which are not martingales in pure sense. In the first part of the paper we will show different types of conditional convergence and relations between them. This article is intended to give a generalization of the theorem about almost sure convergence of martingales and some properties of conditional martingales.