This thesis is concerned with the efficient sequential estimation in continuous time branching processes and in multivariate counting processes. Determining a sequential estimation scheme involves the problem of defining and then finding optimal stopping rules or sampling plans. A lower bound on the variance of an unbiased estimator is given by the (fixed time or sequential) Cramer-Rao type information inequality. If, under a sampling plan S, the equality is attained for an estimator f of a parametric function g = E(f), then S and f are said to be ``efficient`` and g is said to be ``efficiently estimable``. All efficient triples (S,f,g) in a continuous time branching process with immigration with split rate $\lambda_1$, immigration rate $\lambda_2$, offspring distribution [$p_{1j}$, $j\geq0$, $j\neq1$] and immigration distribution [$p_{2j}$, $j\geq1$] are characterized: it is shown that only linear combinations of $\lambda_ip_{ij}$``s and ratios of linear combinations of $\lambda_i^{-1}$ and $\lambda_ip_{ij}$``s to the partial sums of $p_{ij}$``s, i = 1,2, are efficiently estimable when the offspring and immigration distributions are assumed to have finite supports. It is also shown that only the linear combinations of $\mu_i^{-1}$ and $(\lambda_i\mu_i)^{-1}$, the linear combinations of $\mu_i^{-1}$ and $(\lambda_i\mu_i)^{-1}$, and the linear combinations of $\lambda_i^{-1}$ and $\mu_i$ are efficiently estimable when [$p_{ij}$] is of power series type, i = 1,2, where $\mu_1$ and $\mu_2$ are the offspring and immigration means, respectively. Applications to some simple models such as a birth process, a linear birth and death process with immigration and an M/M/$\infty$ queue are also discussed. Efficient sequential estimation in a continuous time branching process without immigration is also considered. Because of the possibility of early, extinction, however, only conditional analysis is considered given either a large initial population or on the set of nonext...