The birth narrative goes back and forth between Matthew and Luke. Each one adding information that the other doesn't.
(Matthew 1:18-25 NIV) This is how the birth of Jesus Christ came about: His mother Mary was pledged to be married to Joseph, but before they came together, she was found to be with child through the Holy Spirit. {19} Because Joseph her husband was a righteous man and did not want to expose her to public disgrace, he had in mind to divorce her quietly. {20} But after he had considered this, an angel of the Lord appeared to him in a dream and said, "Joseph son of David, do not be afraid to take Mary home as your wife, because what is conceived in her is from the Holy Spirit. {21} She will give birth to a son, and you are to give him the name Jesus, because he will save his people from their sins." {22} All this took place to fulfill what the Lord had said through the prophet: {23} "The virgin will be with child and will give birth to a son, and they will call him Immanuel" --which means, "God with us." {24} When Joseph woke up, he did what the angel of the Lord had commanded him and took Mary home as his wife. {25} But he had no union with her until she gave birth to a son. And he gave him the name Jesus.
(Luke 2:1-39 NIV) In those days Caesar Augustus issued a decree that a census should be taken of the entire Roman world. {2} (This was the first census that took place while Quirinius was governor of Syria.) {3} And everyone went to his own town to register. {4} So Joseph also went up from the town of Nazareth in Galilee to Judea, to Bethlehem the town of David, because he belonged to the house and line of David. {5} He went there to register with Mary, who was pledged to be married to him and was expecting a child. {6} While they were there, the time came for the baby to be born, {7} and she gave birth to her firstborn, a son. She wrapped him in cloths and placed him in a manger, because there was no room for them in the inn. {8} And there were shepherds living out in the fields nearby, keeping watch over their flocks at night. {9} An angel of the Lord appeared to them, and the glory of the Lord shone around them, and they were terrified. {10} But the angel said to them, "Do not be afraid. I bring you good news of great joy that will be for all the people. {11} Today in the town of David a Savior has been born to you; he is Christ the Lord. {12} This will be a sign to you: You will find a baby wrapped in cloths and lying in a manger." {13} Suddenly a great company of the heavenly host appeared with the angel, praising God and saying, {14} "Glory to God in the highest, and on earth peace to men on whom his favor rests." {15} When the angels had left them and gone into heaven, the shepherds said to one another, "Let's go to Bethlehem and see this thing that has happened, which the Lord has told us about." {16} So they hurried off and found Mary and Joseph, and the baby, who was lying in the manger. {17} When they had seen him, they spread the word concerning what had been told them about this child, {18} and all who heard it were amazed at what the shepherds said to them. {19} But Mary treasured up all these things and pondered them in her heart. {20} The shepherds returned, glorifying and praising God for all the things they had heard and seen, which were just as they had been told. {21} On the eighth day, when it was time to circumcise him, he was named Jesus, the name the angel had given him before he had been conceived. {22} When the time of their purification according to the Law of Moses had been completed, Joseph and Mary took him to Jerusalem to present him to the Lord {23} (as it is written in the Law of the Lord, "Every firstborn male is to be consecrated to the Lord"), {24} and to offer a sacrifice in keeping with what is said in the Law of the Lord: "a pair of doves or two young pigeons." {25} Now there was a man in Jerusalem called Simeon, who was righteous and devout. He was waiting for the consolation of Israel, and the Holy Spirit was upon him. {26} It had been revealed to him by the Holy Spirit that he would not die before he had seen the Lord's Christ. {27} Moved by the Spirit, he went into the temple courts. When the parents brought in the child Jesus to do for him what the custom of the Law required, {28} Simeon took him in his arms and praised God, saying: {29} "Sovereign Lord, as you have promised, you now dismiss your servant in peace. {30} For my eyes have seen your salvation, {31} which you have prepared in the sight of all people, {32} a light for revelation to the Gentiles and for glory to your people Israel." {33} The child's father and mother marveled at what was said about him. {34} Then Simeon blessed them and said to Mary, his mother: "This child is destined to cause the falling and rising of many in Israel, and to be a sign that will be spoken against, {35} so that the thoughts of many hearts will be revealed. And a sword will pierce your own soul too." {36} There was also a prophetess, Anna, the daughter of Phanuel, of the tribe of Asher. She was very old; she had lived with her husband seven years after her marriage, {37} and then was a widow until she was eighty-four. She never left the temple but worshiped night and day, fasting and praying. {38} Coming up to them at that very moment, she gave thanks to God and spoke about the child to all who were looking forward to the redemption of Jerusalem. {39} When Joseph and Mary had done everything required by the Law of the Lord, they returned to Galilee to their own town of Nazareth.
(Matthew 2 NIV) After Jesus was born in Bethlehem in Judea, during the time of King Herod, Magi from the east came to Jerusalem {2} and asked, "Where is the one who has been born king of the Jews? We saw his star in the east and have come to worship him." {3} When King Herod heard this he was disturbed, and all Jerusalem with him. {4} When he had called together all the people's chief priests and teachers of the law, he asked them where the Christ was to be born. {5} "In Bethlehem in Judea," they replied, "for this is what the prophet has written: {6} "'But you, Bethlehem, in the land of Judah, are by no means least among the rulers of Judah; for out of you will come a ruler who will be the shepherd of my people Israel.'" {7} Then Herod called the Magi secretly and found out from them the exact time the star had appeared. {8} He sent them to Bethlehem and said, "Go and make a careful search for the child. As soon as you find him, report to me, so that I too may go and worship him." {9} After they had heard the king, they went on their way, and the star they had seen in the east went ahead of them until it stopped over the place where the child was. {10} When they saw the star, they were overjoyed. {11} On coming to the house, they saw the child with his mother Mary, and they bowed down and worshiped him. Then they opened their treasures and presented him with gifts of gold and of incense and of myrrh. {12} And having been warned in a dream not to go back to Herod, they returned to their country by another route. {13} When they had gone, an angel of the Lord appeared to Joseph in a dream. "Get up," he said, "take the child and his mother and escape to Egypt. Stay there until I tell you, for Herod is going to search for the child to kill him." {14} So he got up, took the child and his mother during the night and left for Egypt, {15} where he stayed until the death of Herod. And so was fulfilled what the Lord had said through the prophet: "Out of Egypt I called my son." {16} When Herod realized that he had been outwitted by the Magi, he was furious, and he gave orders to kill all the boys in Bethlehem and its vicinity who were two years old and under, in accordance with the time he had learned from the Magi. {17} Then what was said through the prophet Jeremiah was fulfilled: {18} "A voice is heard in Ramah, weeping and great mourning, Rachel weeping for her children and refusing to be comforted, because they are no more." {19} After Herod died, an angel of the Lord appeared in a dream to Joseph in Egypt {20} and said, "Get up, take the child and his mother and go to the land of Israel, for those who were trying to take the child's life are dead." {21} So he got up, took the child and his mother and went to the land of Israel. {22} But when he heard that Archelaus was reigning in Judea in place of his father Herod, he was afraid to go there. Having been warned in a dream, he withdrew to the district of Galilee, {23} and he went and lived in a town called Nazareth. So was fulfilled what was said through the prophets: "He will be called a Nazarene."
(Luke 2:40-52 NIV) And the child grew and became strong; he was filled with wisdom, and the grace of God was upon him. {41} Every year his parents went to Jerusalem for the Feast of the Passover. {42} When he was twelve years old, they went up to the Feast, according to the custom. {43} After the Feast was over, while his parents were returning home, the boy Jesus stayed behind in Jerusalem, but they were unaware of it. {44} Thinking he was in their company, they traveled on for a day. Then they began looking for him among their relatives and friends. {45} When they did not find him, they went back to Jerusalem to look for him. {46} After three days they found him in the temple courts, sitting among the teachers, listening to them and asking them questions. {47} Everyone who heard him was amazed at his understanding and his answers. {48} When his parents saw him, they were astonished. His mother said to him, "Son, why have you treated us like this? Your father and I have been anxiously searching for you." {49} "Why were you searching for me?" he asked. "Didn't you know I had to be in my Father's house?" {50} But they did not understand what he was saying to them. {51} Then he went down to Nazareth with them and was obedient to them. But his mother treasured all these things in her heart. {52} And Jesus grew in wisdom and stature, and in favor with God and men.