Summary: Two new proof techniques for investigating the computational power of two-way computing devices are developed and presented in the thesis. One of them is a completely new technique and the second one is a generalization of the ``cut and paste'' technique originally used in [\textit{A. C. Yao} and \textit{R. L. Rivest}; J. Assoc. Comput. Machin. 25, 337-340 (1978; Zbl 0372.68017)]. Using the first technique we prove that two-way deterministic pushdown automata are more powerful than two-way deterministic counter automata. This method is also used by M. Chrobak in [\textit{M. Chrobak}; J. Comput Syst. Sci. 30, 77-85 (1985; Zbl 0576.68061)] for showing that two-way nondeterministic counter automata are more powerful than two-way deterministic counter automata. These two results settle two open problems posed in [\textit{Z. Galil}; Math. Systems Theory 10(1976), 211-228 (1977; Zbl 0356.68064)]. Our first technique is applied by Z. Galil in [Z. Galil and P. Duris; Inf. Control 54, 217-227 (1982; Zbl 0523.68037)], too, for improving a result of [\textit{T. Chan}; Proc. 13th ACM STOC., 146-157 (1981)]. We define a language L and prove, by virtue of the second proof technique, that any machine that recognizes L must satisfy \(TIME^ 2(n)\cdot Space(n)\geq cn^ 3\). Our machine model allows k to read input heads, where k is fixed, and the movement is like those of a multihead two-way finite automata. This result partially solves an open problem posed in [\textit{A. Borodin} and \textit{S. Cook}; SIAM J. Comput. 11, 287-297 (1982; Zbl 0478.68061), \textit{A. Borodin}, \textit{M. J. Fischer}, \textit{D. G. Kirkpatrick}, \textit{N. A. Lynch} and \textit{M. Tompa}; J. Comput. Syst. Sci. 22, 351-364 (1981; Zbl 0462.68011)]. Partially, since the heads are not allowed to jump. An immediate corollary of this result is that every multihead two-way finite automaton that recognizes L must have a time bound \(T(n)\geq c(n^ 3/\log n)^{1/2}\). This result substantially improves a result of [\textit{L. Janiga}; Fundamentals of computation theory '79, Proc. Conf., 214-218 (1979; Zbl 0413.68085)]. Our second proof technique is the first nondiagonalization method used for establishing the nontrivial time-space lower bound for Turing machines with two read heads on input tape [\textit{A. Borodin}, \textit{M. J. Fischer}, \textit{D. G. Kirkpatrick}, \textit{N. A. Lynch} and \textit{M. Tompa}; J. Comput. Syst. Sci. 22, 351-364 (1981; Zbl 0462.68011)].