A sequence \((H, d)\), \(d^i: H^i\to H^{i+1}\), \(i= 0,\dots, N\) is said to be a quasicomplex if \(d^id^{i-1}\) is compact for all \(i= 1,\dots, N\). If \(H^i= H^{s_i}(X, V^i)\oplus H^{t_i}(Y, W^i)\), \(Y\) is the boundary of \(X\), a smooth compact manifold, \(V^i\), \(W^i\) are vector bundels over \(X\) and \(Y\), \(H^s(X, V)\) is the \(s\)th Sobolev space of sections of \(V\), and \(d_i\) is a \textit{L. Boutet de Monvel} operator [Acta Math. 126, 11--51 (1971; Zbl 0206.39401)], \(i= 0,\dots, N\), this sequence is called a quasicomplex of boundary problems. In this case, sequences of principal interior symbols \(\sigma_\psi(H, d)\); \[ \sigma_\psi(d^{i-1}): \pi^*_X V^{i-1}\to \pi^*_X V^i, \] and boundary symbols \(\sigma_\partial(H, d)\); \[ \sigma_\partial(d^{i-1}): \pi^*_Y((H^{s_{i-1}}(\mathbb{R}_+)\otimes V^{i-1'})\oplus W^{i-1})\to \pi^*_Y((H^{s_i}(\mathbb{R}_+)\otimes V^{i'})\oplus W^i), \] are defined. \((H, d)\) is said to be elliptic, if sequences of principal interior symbols and boundary symbols are both exact. In this paper, the existence of elliptic complexes having the same sequence of principal symbol for any elliptic quasicomplex of boundary value problems is proved (Th. 8.1). Since quasicomplexes are stable under compact perturbation, this result allows to define Euler characteristics of elliptic quasicomplexes (Def. 8.1 and Th. 8.2). The authors say that the construction of the elliptic complex having the same principal symbol sequences of a given elliptic quasicomplex was tried in [\textit{U. Pillat} and \textit{B.-W. Schulze}, Math. Nachr. 94, 173--210 (1980; Zbl 0444.58017)]. But the proof was wrong and this paper gives a correct proof for the first time. Besides this result, several results on general properties of quasicomplexes and their applications to boundary problems are also presented. The outline of the paper is as follows: The first two sections (\S1 and \S2) review Boutet Monvel calculus and quasicomplex. Elliptic quasicomplex is defined in \S3. An elliptic quasicomplex is Fredholm, that is, transformed to a complex by a compact perturbation, if \(s\) is large. This is proved in \S4 (Th.4.5). Since a quasicomplex \((H, d)\) of Hilbert spaces is Fredholm if and only if all Laplacians \(\Delta^i= d^{i-1} d^{i-1*}+ d^{i*} d^i\) are Fredholm (Lemma 5.1), Hodge theory for elliptic quasicomplexes is possible. This is described in \S5 together with the construction of the Green operator (parametrix) of \(\Delta^i\) (Th.5.2 and 5.4). Let \((L, a)\) and \((M, bn)\) be quasicomplexes. Then a map \(t: (L, a)\to (M, b)\); \(t^i: L^i\to M^i\), such that \(t^{i+1} a^i- b^i t^i\) is compact, \(i= 0,\dots, N\), is called a quasicomplex mapping. Then the complex \((L^i\oplus M^{i-1}, d^i)\), where \[ d^i= \begin{pmatrix} -a^i & 0\\ t^i & b^{i-1}\end{pmatrix}, \] is called the cone of the quasicochain mapping \(t\) and denoted by \({\mathcal C}(t)\) (\S6). If \(L= H^s(X, V)\), \(M= H^l(Y, W)\), and the cone is elliptic, it is called boundary problems for elliptic complexes of pseudo differential operators on \(X\) [\textit{A. Dynin}, Funct. Anal. Appl. 6, 67--68 (1972); translation from Funkts. Anal. Prilozh. 6, 75--76 (1972; Zbl 0266.35063)]. \S7 deals with application of cone to overdetermined boundary problems. Existence of complexes having the same principal symbol sequence of a given elliptic quasicomplex is proved in \S8 by using Hodge theory described in \S5. The existence of these complexes provides the definition of Euler characteristic class of an elliptic quasicomplex (Def. 8.1 and Th. 8.2). The authors expect this will serve to index theory of elliptic quasicomplexes. Taking \(V^i= V\oplus\wedge^i T^*\) and \(d^i= d+ A\), \(A\) is a connection of \(V\), we obtain a quasicomplex. Motivated by this example, the connection of a quasicomplex and its curvature are defined in \S9, the last section. But the authors say, detailed study on this topic exceeds the scope of this paper.