Theophanies (manifestations of God) light up the way of the promise, from the patriarchs to Moses and from Joshua to the visions that inaugurated the missions of the great prophets. Christian tradition has always recognized that God's Word allowed himself to be seen and heard in these theophanies, in which the cloud of the Holy Spirit both revealed him and concealed him in its shadow.--CCC §707

Genesis 12--And YHWH appeared to Abram, and He said, "To your seed I will give this land," and there he built an altar to YHWH, Who had appeared to him.

It seems uncertain exactly here how God appeared to Abraham, this verse is also quoted by St Paul, peace be up him, for Christ by the use the singular word "seed" for Christ.

Genesis 18--"Now YHWH appeared to him in the plains of Mamre and he was sitting at the entrance of the tent when the day was hot.  And he lifted his eyes and saw, and behold, three men were standing beside him, and he saw and he ran toward them from the entrance of the tent, and he prostrated himself to the ground. And he said, "My lords [others read lord, Adonai in Hebrew], if only I have found favor in your eyes, please do not pass on from beside your servant."...And YHWH said, "Since the cry of Sodom and Gomorrah has become great, and since their sin has become very grave, I will descend now and see, whether according to her cry, which has come to Me, they have done; [I will wreak] destruction [upon them]; and if not, I will know."   And the men turned from there and went to Sodom, and Abraham was still standing before YHWH. 

Here it is uncertain whether the 3 men were Angels of the Lord, or the LORD Himself as a theopany, the text seems to call atleast one of them Lord, Genesis 19:1 calls two of the "angels."  The text is ambiguous since the name YHWH was not revealed to Man until Moses, peace be upon him, so this explains why the text refers to LORD, but Abraham himself uses adonai, which can be used to either man or God.

"he calleth them lords, without knowing who they are who are come to him, and presents his wife in the place of a handmaiden."--St John Chrysostom Homily 35, on 1 Cor. 14

"Again with the reverence due to an excellent creature Abraham adored the angels, as did also Josue [Joshua 5:15]: though we may understand them to have adored, with the adoration of latria, God Who appeared and spoke to them in the guise of an angel.....Since there is one excellence of the three Divine Persons, one honor and reverence is due to them and consequently one adoration. It is to represent this that where it is related [Genesis 18:2] that three men appeared to Abraham, we are told that he addressed one, saying: "Lord, if I have found favor in thy sight," etc. The triple genuflection represents the Trinity of Persons, not a difference of adoration."--St Thomas Aquinas, Summa Theologicae 2:2:Q84

Ver. 2. Men in outward appearance, but angels indeed. (Hebrews xiii. 2; St. Augustine, City of God xvi. chap. 29.) Some have supposed, that one of them was the Son of God, whom Abraham adored, and who bears throughout the chief authority. Tres vidit et unum adoravit. He saw three and adored one, as we read in the Church office. In the former supposition, which is generally adopted, this adoration was only a civil ceremony, if Abraham considered them as mere men; or it might be mixed with a degree of religious, though inferior veneration, if he imagined they were angels; or in fine, he adored God in his representatives. (Haydock)

The text refers to the three as "men," then later two of them are called "men."  "I will descend now and see"....here is almost appears as if God were saying He was not omniscient, the purpose is possibly simply to see if the people of Sodom will sin against God in form of men or angels.

Genesis 32:13--"And Jacob was left alone, and a man wrestled with him until the break of dawn."

Disputed Theopany:

Joshua 5:13-5, 6:2--And it was when Joshua was in Jericho, that he lifted up his eyes and saw, and, behold, a man was standing opposite him with his sword drawn in his hand; and Joshua went to him, and said to him, "Are you for us, or for our adversaries?" And he said, "No, but I am the the captain of the host of YHWH; I have now come."  And Joshua fell on his face to the earth and prostrated [most bibles read worshipping here] himself, and said to him, "What does my lord [Heb. adonai] say to his servant?" And the captain of YHWH's host said to Joshua, "Remove your shoe from your foot; for the place upon which you stand is holy."  And Joshua did so....And the Lord said to Joshua, See, I have given into your hand Jericho and its king, the mighty warriors.

""I shall give you another testimony, my friends," said I, "from the Scriptures, that God begat before all creatures a Beginning, [who was] a certain rational power [proceeding] from Himself, who is called by the Holy Spirit, now the Glory of the Lord, now the Son, again Wisdom, again an Angel, then God, and then Lord and Logos; and on another occasion He calls Himself Captain, when He appeared in human form to Joshua the son of Nave (Nun). For He can be called by all those names, since He ministers to the Father's will, and since He was begotten of the Father by an act of will; just as we see happening among ourselves.....Listen, therefore, to the following from the book of Joshua, that what I say may become manifest to you; it is this: 'And it came to pass, when Joshua was near Jericho, he lifted up his eyes, and sees a man standing over against him. And Joshua approached to Him, and said, Are you for us, or for our adversaries? And He said to him, I am Captain of the Lord's host: now have I come. And Joshua fell on his face on the ground, and said to Him, Lord, what do You command Your servant? And the Lord's Captain says to Joshua, Loose the shoes off your feet; for the place whereon you stand is holy ground. And Jericho was shut up and fortified, and no one went out of it. And the Lord said to Joshua, Behold, I give into your hand Jericho, and its king, [and] its mighty men.' """--St Justin the Martyr c AD 110 Dialogue with Trypho Chapter 61

Ver. 13. Prince of the host of the Lord, &c. St. Michael, who is called prince of the people of Israel, Daniel x. 21. (Challoner) --- Some of the Fathers explain it of the son of God. (Origen, hom. 6.) But St. Augustine, City of God xi. 13., St. Jerome in Galatians iii., and interpreters in general agree, that the person who here appeared to Josue, was the archangel Michael. He came, in the name of God, to assure Josue of success, as the angel had appeared to Moses in the burning bush, as if to denote the distress of the Hebrews, and to encourage Moses to undertake their liberation. (Calmet) --- Chaldean, "I am the angel sent by God." In that character he is called the Lord. (Haydock)

Ver. 14. Worshipping. Not with divine honour, but with a religious veneration of an inferior kind, suitable to the dignity of his person. (Challoner) --- He styles the angel Adonai, which is a title frequently given to men; and hence he does not seem to have designed to give him supreme worship. (Calmet) --- If he did, (Haydock) it was referred to God. (Calmet) (Exodus xx.)

Ver. 15. Loose. The angel did not only accept of the honour done to him, but also required more, shewing that the field near Jericho was rendered holy, by his presence. (Worthington) --- Hence he ordered Josue to put off his shoes, as Moses had done at the bush, Exodus iii. 5. The Turks leave their shoes at the doors of their mosques, and do not dare to tread on the bare floor. Formerly the pagans would not spit in their temples. (Arrian.) "If, says Porphyrius, in the sacrifices instituted by men, in honour of the gods, people be careful to have their shoes clean, with how much greater attention ought we to preserve our bodies, which are, as it were, the garments of the soul, free from every impurity and corruption!" (Abstin. 2.) (Calmet)

"the captain of YHWH’s host " Israel, who is the Lord’s host. Now, this was Michael, as it is said: “Michael your prince."--Rashi, Jewish Commentary

Rejected Theopany

Melchizedek