What exactly do you mean by random variable converge almost surely?

Question

If somebody claim that $\sum_{k=1}^n X_k$ converges almost surely to a random variable , does it imply that $Y$ is finite? If so, first of all, is it the point of proving Kolmogorov convergence theorem and Kolmogorov 3 three series theorem being trying to $\sum_{k=1}^n X_k$ is finite? And, providing that $X_k$ is nonnegative, is it true that it only suffices to check if $\sum_{k=1}^n X_k<\infty$?