The notion of Bochner (i.e., uniform) almost periodicity for a function $y \: \Bbb{R} \to X$ taking values in a real Banach space $X$ may be generalized in various ways. For example, replacing the uniform convergence by the pointwise convergence, one obtains the Banach space $\text{AA}(\Bbb{R},X)$ of almost automorphic functions, with the supremum norm $\|y\|_{\infty}$. On the other hand, if the interest is in the behavior at infinity, then it is natural to consider functions of the form $y + \phi$ with $y \in \text{AA}(\Bbb{R},X)$ and $\phi \in C_0(\Bbb{R}_+,X)$ vanishing at infinity. This leads to the Banach space $\text{AAA}(\Bbb{R}_+,X)$ of asymptotically almost automorphic functions, with norm $\|y\|_{\infty} + \|\phi\|_{\infty}$. \par The authors obtain results on the existence and uniqueness of solutions in $\text{AA}(\Bbb{R},X)$ and $\text{AAA}(\Bbb{R}_+,X)$, respectively, for two classes of non-autonomous non-linear abstract integral equations. In their general form, these equations are of mixed type, i.e., the present value of the unknown depends on both its past and future values. The authors also study the existence and uniqueness of mild solutions in $\text{AAA}(\Bbb{R}_+,X)$ for a related class of abstract integro-differential equations with non-local initial conditions. In all cases the word `abstract' reflects that the unknown is allowed to take values in an arbitrary Banach space. \par So far, I have restricted my attention to the discussion of the almost automorphic case considered in Sections 2, 3, and 4, centering around the equation $$ \multline y(t) = f(t,y(t),y(a_0(t))) + \int_{-\infty}^t C_1(t,s,y(s),y(a_1(s)))\,ds\\ + \int_t^{\infty} C_2(t,s,y(s),y(a_2(s)))\,ds, \quad t \in \Bbb{R}, \endmultline \tag1 $$ where $f \: \Bbb{R} \times X \times X \to X$, $C_{1},C_{2} \: \Bbb{R} \times \Bbb{R} \times X \times X \to X$, and $a_{0},a_{1} \: \Bbb{R} \to \Bbb{R}$ satisfy certain conditions. The text is accessible and stimulates thinking about the existing work as well as future directions. The remarks and questions that follow below are a consequence of these qualities and are meant to be constructive. \par In Section 2, the definition of bi-almost automorphicity of $C \: \Bbb{R} \times \Bbb{R} \times X \times X \to X$ with respect to $(s,t) \in \Bbb{R} \times \Bbb{R}$, uniformly for $(x,y)$ in bounded subsets of $X \times X$, is recalled from the literature and the associated limit function $\tilde{C}$ is defined. The authors say that $C$ is $\lambda$-bounded if $C$ is continuous and for any bounded set $\Cal{B} \subseteq X \times X$ there exists $\lambda \: \Bbb{R} \times \Bbb{R} \to \Bbb{R}_+$ such that $$ \|C(t+\tau,s+\tau,x,y)\| \le \lambda(t,s) \quad \text{for all } (t,s,\tau) \in \Bbb{R} \times \Bbb{R} \times \Bbb{R}, $$ and all $(x,y) \in \Cal{B}$. Both notions play an important role in the rest of the article, but the subsequent Lemma 2.9 in its printed form is not strong enough for application to, e.g.,\ the last step in the proof of Lemma 2.13 and the estimates of the terms $\Lambda_{i,n}$ in the proof of Theorem 4.3. Fortunately, the corresponding author A.\ Chávez has kindly and quickly communicated a stronger formulation to me. Specifically, he shows that if $C$ is bi-almost automorphic and $\lambda$-bounded, then the limit function $\tilde{C}$ satisfies the same $\lambda$-bound. In this sense the definition of $\lambda$-boundedness is actually symmetric. \par It is of some importance to keep in mind that the various definitions and technical conditions introduced in Section 2 typically depend on the selection of a specific subsequence, a bounded subset, or both. This dependence is not reflected in the notation, and, while in most cases it is clear from the context, sometimes the reader is left guessing. For example, in condition $(K_2)$ preceding Theorem 3.5, it is not obvious whether, or how, the constants $N_i$ and $\alpha_i$ introduced in conditions (H3) and (H4) in Section 2.2 are supposed to depend on the given $r$-ball, but this question is actually relevant for an appreciation of the theorem and its proof. \par In Section 3, Banach's fixed point theorem is used to prove the sufficiency of a number of related conditions for the existence of the unique solutions, in $\text{AA}(\Bbb{R},X)$, of (1) and its special case when $C_2 = 0$. \par In Section 4, in Theorem 4.3 a result of (Massera-)Bohr-Neugebauer type is presented, establishing the sufficient condition in the statement that a bounded and continuous solution $y$ of (1) is in $\text{AA}(\Bbb{R},X)$ if and only if the range of $y$ is relatively compact. I find these sorts of theorems very appealing. \par \{Reviewer's remarks: I believe that in assumption 1 of Theorem 3.1, it would be better to formulate in terms of vectors in $X$, requiring existence of $L_f > 0$ such that for all $t \in \Bbb{R}$ and for all $v,w,\widehat{v},\widehat{w}$ in the closed $\rho$-ball centered at the origin in $X$, it holds that $$ \|f(t,v,w) - f(t,\widehat{v},\widehat{w})\| \le L_f(\|v - \widehat{v}\| + \|w - \widehat{w}\|). $$ Otherwise, it is not clear to me how the authors are able to use Lemma 2.12 (or, rather, its almost automorphic analogue) to deduce that $f(\cdot,y(\cdot),y(a_0(\cdot)))$ in the fixed point operator for (1) is in $\text{AA}(\Bbb{R},X)$ and to make the estimates for $f$ that appear twice in the proof. Similar comments apply to Theorems 3.3 and 3.5. \par \{Theorem 4.3 may need some clarification. Here my concern is with the construction of the functions $\tilde{y} \: \Bbb{R} \to X$ and $\tilde{y}_i \: \Bbb{R} \to X$ at the beginning of the proof. Each of these is introduced via a pointwise selection of convergent subsequences. The complication is that, potentially, not just for fixed $t$ but in fact for every $s \in \Bbb{R}$ a further subsequence may need to be selected. Indeed, let $\{u_n\}$ be a sequence of functions on $\Bbb{R}$ into a metric space such that there exists a compact set $K$ with $u_n(s) \in K$ for all $n \in \Bbb{N}$ and all $s \in \Bbb{R}$. Then it need not be true that $\{u_n\}$ admits a pointwise convergent subsequence, even if we replace $\Bbb{R}$ by a compact interval, choose $\Bbb{R}_+$ for the metric space, and assume each $u_n$ to be continuous. The same complication appears in the construction of the auxiliary function $\eta \: \Bbb{R} \to \Bbb{R}_+$ at the end of the proof.\}